Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy

Multiple orthogonal p olynomials of mixed t yp e: Gauss-Borel factorization and the m ulti-comp onen t 2D T o da hierarc h y Carlos ´ Alv arez-F ern´ andez 1 , 3 , † , Ulises Fidalgo Prieto 2 , ‡ and Man uel Ma ˜ nas 1 , z 1 Departamen to de F ´ ısica T e´ orica II, Universidad Complutense, 28040-Madrid, Spain 2 Departamen to de Matem´ atica Aplicada, Univ ersidad Carlos I I I 28911-Madrid, Spain 3 Departamen to de M´ eto dos Cuan titativ os, Universidad P on tiﬁcia Comillas, 28015-Madrid, Spain † calv arez@cee.upcomillas.es, ‡ uﬁdalgo@math.uc3m.es, z man uel.manas@ﬁs.ucm.es Abstract Multiple orthogonality is considered in the realm of a Gauss–Borel factorization problem for a semi-inﬁnite momen t matrix. Perfect com binations of weigh ts and a ﬁnite Borel measure are constructed in terms of M-Nikishin systems. These p erfect com binations ensure that the problem of mixed m ultiple orthogonalit y has a unique solution, that can b e obtained from the solution of a Gauss–Borel factorization problem for a semi-inﬁnite matrix, whic h pla ys the role of a moment matrix. This leads to sequences of multiple orthogonal p olynomials, their duals and second kind functions. It also gives the corresp onding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these ob jects in terms of determinan ts from the moment matrix are given, recursion relations are found, which imply a m ulti-diagonal Jacobi t yp e matrix with snake shap e, and results lik e the ABC theorem or the Christoﬀel–Darb oux formula are re-derived in this con text (using the factorization problem and the generalized Hank el symmetry of the momen t matrix). The connection b et ween this description of multiple orthogonality and the multi-component 2D T oda hierarc hy , which can be also understo od and studied through a Gauss–Borel factorization problem, is discussed. Deformations of the weigh ts, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarc hy , represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov–Shabat matrices as w ell as wa ve functions and their adjoints are determined. The construction of discrete ﬂo ws is discussed in terms of Miw a transformations whic h in v olve Darboux transformations for the m ultiple orthogonality conditions. The bilinear equations are derived and the τ -function represen tation of the multiple orthogonality is giv en. 1 Con ten ts 1 In tro duction 2 1.1 The Gauss–Borel factorization of the moment matrix and orthogonal p olynomials . . . . . . . . . . . . 4 2 Multiple orthogonal polynomials and Gauss–Borel factorization 7 2.1 The moment matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 P erfect combinations and Nikishin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 The inv erse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 The Gauss–Borel factorization and multiple orthogonal p olynomials . . . . . . . . . . . . . . . . . . . 14 2.4 Linear forms and multiple bi-orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 F unctions of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Recursion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Christoﬀel–Darb oux t yp e formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7.1 Pro jection op erators and the Christoﬀel–Darb oux k ernel . . . . . . . . . . . . . . . . . . . . . . 24 2.7.2 The ABC type theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7.3 Christoﬀel–Darb oux form ula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Connection with the multi-component 2D T o da Lattice hierarc h y 30 3.1 Con tinuous deformations of the momen t matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Lax equations and the integrable hierarch y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Darb oux–Miw a discrete ﬂows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.1 Miw a transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 Bounded from b elo w measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Symmetries, recursion relations and string equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Bilinear equations and τ -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 App endices 44 A Pro ofs 44 B Discrete ﬂo ws asso ciated with binary Darb oux transformations 51 1 In tro duction The topic of m ultiple orthogonalit y of p olynomials is very close to that of simultaneous rational approximation (si- m ultaneous Pad ´ e apro ximan ts) of systems of Cauch y transforms of measures. The history of sim ultaneous rational appro ximation starts in 1873 with the well known article [22] in which Ch. Hermite prov ed the transcendence of the Euler num b er e. Later, around the years 1934-35, K. Mahler delivered at the Universit y of G roningen several lectures [27] where he settled do wn the foundations of this theory . Mean while, tw o of Malher’s students, J. Coates and H. Jager, made imp ortan t con tributions in this resp ect (see [13] and [23]). In the case of Cauc hy transforms, the simultaneous rational approximation deﬁnition may b e written in terms of multiple orthogonality of p olynomials as follo ws. Giv en an in terv al ∆ ⊂ R of the real line, let M (∆) denote all the ﬁnite Borel measures which hav e supp ort, supp( · ) with inﬁnitely many p oin ts in ∆ , where they do not change sign. Fix µ ∈ M (∆), and let us consider a system of weigh ts ~ w = ( w 1 , . . . , w p ) on ∆, with p ∈ N . (In this pap er a“weigh t” on an in terv al ∆ is mean t to b e a real in tegrable function deﬁned on ∆ which do es not c hange its sign on ∆.) Fix a m ulti-index ~ ν = ( ν 1 , . . . , ν p ) ∈ Z p + , Z + = { 0 , 1 , 2 , . . . } , and denote | ~ ν | = ν 1 + · · · + ν p . There exist p olynomials, A 1 , . . . , A p , not all iden tically equal to zero which satisfy the follo wing orthogonality relations Z ∆ x j p X a =1 A a ( x ) w a ( x )d µ ( x ) = 0 , deg A a ≤ ν a − 1 , j = 0 , . . . , | ~ ν | − 2 . (1) Analogously , there exists a p olynomial B not iden tically equal to zero, such that Z ∆ x j B ( x ) w b ( x )d µ ( x ) = 0 , deg B ≤ | ~ ν | , j = 0 , . . . , ν b − 1 , b = 1 , . . . , p. (2) 2 The resulting polynomials are said to b e of t yp e I and type II, respectively , with resp ect to the com bination ( µ, ~ w , ~ ν ) of the measure µ , the systems of weigh ts ~ w and the multi-index ~ ν . When p = 1 b oth deﬁnitions coincide with that of the standard orthogonal p olynomials on the real line. The existence of a system of polynomials ( A 1 , . . . , A p ) and a p olynomial B deﬁned from (1) and (2) resp ectiv ely , are ensured because in b oth cases ﬁnding the co eﬃcien ts of the p olynomials is equiv alen t to solving a system of | ~ ν | linear homogeneous equations with | ~ ν | + 1 unkno wn coeﬃcients. F rom the theory of orthogonal p olynomials w e kno w that when p = 1 eac h polynomial A 1 ≡ B has exactly degree | ~ ν | = ν 1 ; unfortunately if p > 1 that is not true in general. F or instance, let us take a system of w eigh ts ~ w = ( w 1 , w 1 , . . . , w 1 ), in this case the solution vector space has dimension bigger than one, and we can ﬁnd t wo solutions which are linearly indep enden t. Hence, there is at least an a ∈ { 1 , . . . , p } such that deg A a < ν a − 1 and deg B < | ~ ν | . Given a measure µ ∈ M (∆) and a system of w eights ~ w on ∆ a multi-index ~ ν is called type I or type II normal if deg A a m ust equal to ν a − 1 , a = 1 , . . . , p , or deg B m ust equal to | ~ ν | − 1, resp ectiv ely . When for a pair ( µ, ~ w ) all the multi-indices are type I or t yp e I I normal, then the pair is called type I p erfect or t yp e I I p erfect, resp ectiv ely . The concepts of normality and p erfectness were introduced by Malher (see Malher’s, Coates’ and Jager’s articles cited ab ov e). Multiple orthogonal of p olynomials hav e b een employ ed in sev eral pro ofs of irrationality of num b ers. F or example, in [10], F. Beukers shows that Ap ery’s pro of (see [8]) of the irrationalit y of ζ (3) can b e placed in the context of a com bination of t yp e I and t ype I I m ultiple orthogonality , which is called mixed type multiple orthogonality of p olynomials. More recently , mixed t yp e approximation has app eared in random matrix and non-in tersecting Bro wnian motion theories (see, for example, [11], [14] and [25]). A formalization of this kind of orthogonality w as initiated by V. N. Sorokin [36]. He studied a sim ultaneous rational appro ximation construction which is closely connected with m ultiple orthogonal p olynomials of mixed type. Surprisingly , in [21] a Riemann–Hilb ert problem was found for the theory of orthogonal polynomials, and later [39] this result w as largely extended to type I and I I multiple orthogonality . In [14] mixed type multiple orthogonality was analyzed from this p ersp ectiv e. In order to introduce multiple orthogonal p olynomials of mixed type we need t wo systems of weigh ts ~ w 1 = ( w 1 , 1 , . . . , w 1 ,p 1 ) and ~ w 2 = ( w 2 , 1 , . . . , w 2 ,p 2 ) where p 1 , p 2 ∈ N , (as w e said a set of functions whic h do not c hange their sign in ∆), and tw o m ulti-indices ~ ν 1 = ( ν 1 , 1 , . . . , ν 1 ,p 1 ) ∈ Z p 1 + and ~ ν 2 = ( ν 2 , 1 , . . . , ν 2 ,p 2 ) ∈ Z p 2 + with | ~ ν 1 | = | ~ ν 2 | + 1. There exist polynomials A 1 , . . . , A p 1 , not all iden tically zero, such that deg A s < ν 1 ,s whic h satisfy the follo wing relations Z ∆ p 1 X a =1 A a ( x ) w 1 ,a ( x ) w 2 ,b ( x ) x j dµ ( x ) = 0 , j = 0 , . . . , ν 2 ,b − 1 , b = 1 , . . . , p 2 . (3) They are called mixed multiple-orthogonal p olynomials with resp ect to the combination ( µ, ~ w 1 , ~ w 2 , ~ ν 1 , ~ ν 2 ) of the measure µ, the systems of w eights ~ w 1 and ~ w 2 and the multi-indices ~ ν 1 and ~ ν 2 . It is easy to sho w that ﬁnding the p olynomials A 1 , . . . , A p 1 is equiv alent to solving a system of | ~ ν 2 | homogeneous linear equations for the | ~ ν 1 | unknown co eﬃcien ts of the polynomials. Since | ~ ν 1 | = | ~ ν 2 | + 1 the system alw ays has a nontrivial solution. The matrix of this system of equations is the so called momen t matrix, and the study of its Gauss–Borel factorization will b e the cornerstone of this pap er. Observ e that when p 1 = 1 w e are in the type II case and if p 2 = 1 in type I case. Hence in general we can ﬁnd a solution of (3) where there is an a ∈ { 1 , . . . , p 1 } such that deg A a < ν 1 ,a − 1 . When giv en a com bination ( µ, ~ w 1 , ~ w 2 ) of a measure µ ∈ M (∆) and systems of weigh ts ~ w 1 and ~ w 2 on ∆ if for each pair of multi- indices ( ~ ν 1 , ~ ν 2 ) the conditions (3) determine that deg A a = ν 1 ,a − 1 , a = 1 , . . . , p 1 , then we sa y that the com bination ( µ, ~ w 1 , ~ w 2 ) is p erfect. The concept of p erfe ctness will be rigorously in tro duced in Deﬁnition 2. The seminal pap er of M. Sato [33], and further dev elopmen ts performed b y the Kyoto school through the use of the bilinear equation and the τ -function formalism [16]-[18], settled the basis for the Lie group theoretical description of in tegrable hierarchies, in this direction we hav e the relev ant contribution b y M. Mulase [30] in which the factor- ization problems, dressing procedure, and linear systems were the k ey for in tegrability . In this dressing setting the m ulticomp onen t in tegrable hierarchies of T oda type were analyzed in depth b y K. Ueno and T. T ak asaki [37]. See also the pap ers [9] and [24] on the multi-component KP hierarch y and [28] on the m ulti-component T oda lattice hierarc hy . In a series of pap ers M. Adler and P . v an Mo erbeke sho wed ho w the Gauss–Borel factorization problem app ears in the theory of the 2D T oda hierarc h y and what they called the discrete KP hierarc h y [1]-[5]. In these pap ers it b ecomes clear –from a group-theoretical setup– why standard orthogonality of p olynomials and in tegrability of nonlinear equations of T o da type where so close. In fact, the Gauss–Borel factorization of the moment matrix ma y b e understo od as the Gauss–Borel factorization of the initial condition for the integrable hierarch y . T o see the connection b et ween the work of Mulase and that of Adler and v an Moerb eke see [19]. Later on, in the recen t pap er [6], it is sho wn that the multiple orthogonal construction describ ed in previous paragraphs was linked with the multi- comp onen t KP hierarch y . In fact, for a giv en set of weigh ts ( ~ w 1 , ~ w 2 ) and degrees ( ~ ν 1 , ~ ν 2 ) the authors constructed a ﬁnite matrix that pla ys the role of the moment matrix and, using the Riemann-Hilbert problem of [14], where able to sho w that determinan ts constructed from the moment matrix were τ -functions solving the bilinear equation for the m ulti-comp onen t KP hierarch y . How ever, there is no men tion in that pap er to an y Gauss–Borel factorization in spite 3 of b eing the multicomponent in tegrable hierarchies connected with diﬀeren t factorization problems of these type. F or further developmen ts on the Gauss–Borel factorization and multi-component 2D T o da hierarch y see [7] and [29]. This motiv ated our initial researc h in relation with this pap er; i.e., the construction of an appropriate Gauss– Borel factorization in the group of semi-inﬁnite matrices leading to multiple orthogonalit y and in tegrability in a sim ultaneous manner. The main adv antage of this approach lies in the application of diﬀerent tec hniques based on the factorization problem used frequen tly in the theory of in tegrable systems. The key ﬁnding of this pap er is, therefore, the characterization of a semi-inﬁnite moment matrix whose Gauss–Borel factorization leads directly to multiple orthogonalit y . This makes sense when factorization can b e p erformed, whic h is the case for p erfect combinations ( µ, ~ w 1 , ~ w 2 ), which allo ws us to consider some sets of multiple orthogonal polynomials (called ladders) very m uch in the same manner as in the (non multiple) orthogonal p olynomial setting. The Gauss–Borel factorization of this momen t matrix leads, when one takes into account the Hankel type symmetry of the moment matrix, to results like: 1. Recursion relations, 2. ABC theorems and 3. Christoﬀel–Darb oux formulas. The ﬁrst tw o are new results while the third is not new, as it w as derived from the Riemann–Hilb ert problem in [14]. Ho wev er, our deriv ation of the Christoﬀel–Darb oux formula is based exclusiv ely on the Gauss–Borel factorization, and its uniqueness and existence for the m ultiple orthogonalit y problem are the only requirements. Th us, it is suﬃcient to hav e a p erfect combination ( µ, ~ w 1 , ~ w 2 ), and there are examples of this t yp e which do not hav e a w ell deﬁned Riemann–Hilb ert problem in the spirit of [14]. When we seek for the appropriate integrable hierarc hy link ed with multiple orthogonalit y w e are lead to the m ulticomp onen t 2D T o da lattice hierarch y whic h extends the construction of the multicomponent KP hierarch y considered by M. J. Bergv elt and A. P . E. ten Kro ode in [9]; not to the multicomponent 2D T o da lattice hierarch y as describ ed in [37] or [28]. In the spirit of this last men tioned articles, and complemen ting the contin uous ﬂows of the in tegrable hierarch y , w e also introduce discrete ﬂo ws, that could b e view ed as Darb oux transformations, and whic h corresp ond to Miwa transformations implying the addition of a zero/p ole to the set of weigh ts. Moreo ver, the Hank el t yp e symmetry is related to an inv ariance under a n umber of ﬂows, and to string equations. Bilinear equations can b e deriv ed from the Gauss–Borel factorization problem and moreov er the τ -function represen tation is a v ailable leading to a bridge to the results of [6] in whic h no semi-inﬁnite matrix or Gauss–Borel factorization was used. This pap er is divided into three sections, § 1 is this in tro duction which con tains § 1.1 in where we review the application of the LU factorization of the moment matrix to the theory of orthogonal polynomials in the real line. Next, § 2 is devoted to the presen tation of the moment matrix and the discussion of the Gauss–Borel factorization. In this form we obtain perfect systems in terms of Nikishin systems, determinantal expressions for the m ultiple orthogonal p olynomials, their duals and second type functions, bi-orthogonality for the asso ciated linear forms, recursion re lations, ABC type theore ms and the Christoﬀel–Darb oux formula. Flows and the integrable hierarch y are studied in § 3 in whic h an in tegrable hierarch y a la Bergvelt-ten Kro ode is link ed with the multiple orthogonalit y problem. W e not only derive from the Gauss–Borel factorization the Lax and Zak arov–Shabat equations, but also w e introduce discrete in tegrable ﬂo ws, describ ed by Miwa shifts, or Darb oux transformations, and also construct an appropriate bilinear equation. Finally , w e ﬁnd the τ functions corresp onding to the m ultiple orthogonality and link them to those of [6]. A t the end of the pap er, w e ha ve added tw o app endices: the ﬁrst one collects the more tec hnical pro ofs of the results in this pap er. In Appendix B w e consider discrete ﬂows for the case of a measure µ with un b ounded supp ort supp µ . 1.1 The Gauss–Borel factorization of the momen t matrix and orthogonal polynomials Here we discuss ho w the LU factorization of the standard moment matrix g = ( R x i + j d µ ) of a constant sign ﬁnite Borel measure µ leads to traditional results in the theory of orthogonal p olynomials, namely recursion relation and Christoﬀel–Darb oux form ula. In spite that these results are well established we rep eat them here b ecause in their deriv ation is enco ded the set of argumen ts w e will use in the multiple orthogonalit y setting. In the forthcoming exp osition it will b ecome clear the LU factorization approac h is just a compact wa y of using the orthogonality relations. The moment matrix can b e written as the follo wing Grammian matrix g = Z χ ( x ) χ ( x ) > d µ ( x ) in terms of the monomial string χ ( x ) := (1 , x, x 2 , . . . ) > . The Borel–Gauss factorization of g is g = S − 1 ¯ S , S =   1 0 0 ··· S 1 , 0 1 0 ··· S 2 , 0 S 2 , 1 1 ··· . . . . . . . . . . . .   , ¯ S − 1 =    ¯ S 0 0 , 0 ¯ S 0 0 , 1 ¯ S 0 0 , 2 ··· 0 ¯ S 0 1 , 1 ¯ S 0 1 , 2 ··· 0 0 ¯ S 0 2 , 2 ··· . . . . . . . . . . . .    . 4 The reader should notice that • It makes sense whenever the truncated momen t matrix g [ l ] = ( g i,j ) 0 ≤ i,j , ¯ P := ( ¯ S − 1 ) > χ = ( ¯ P 0 , ¯ P 1 , . . . ) > . The families of p olynomials { P l } ∞ l =0 and { ¯ P k } ∞ k =0 are biorthogonal: Z P ( x ) ¯ P ( x ) > d µ ( x ) = Z S χ ( x ) χ ( x ) > ¯ S − 1 d µ ( x ) = S Z χ ( x ) χ ( x ) > d µ ( x ) ¯ S − 1 = I ⇒ Z P l ( x ) ¯ P k ( x )d µ ( x ) = δ l,k . In this simple pro of relies the basic connection b et ween orthogonality and the LU factorization, which we consider as the very same thing dressed in diﬀerent manners. F rom the ab o ve orthogonality we conclude that Z P l ( x ) x j d µ ( x ) = 0 , j = 0 , . . . , l − 1 , Z ¯ P l ( x ) x j d µ ( x ) = 0 , j = 0 , . . . , l − 1 , (4) and we also hav e that P l ( x ) and ¯ P l are l -th degree p olynomials where P l is monic and ¯ P l satisﬁes R x l ¯ P l ( x )d µ ( x ) = 1, i.e. w e hav e type I I and type I normalizations. Given that the momen t matrix is symmetric, g = g > and the uniqueness of the LU factorization we deduce that ¯ S = H ( S − 1 ) > , with H = diag( h 0 , h 1 , . . . ); i.e., g = S − 1 H ( S − 1 ) > and the factorization is a Cholesky factorization (but this do es not extend to the multiple orthogonal case). Therefore ¯ P l = h − 1 l P l so that R P l ( x ) P k ( x )d µ ( x ) = h l δ l,k , and { P l } ∞ l =0 is a family of monic orthogonal p olynomials with resp ect to the measure µ . Considering the orthogonality relations as a linear system for the co eﬃcien ts of the p olynomials one concludes that p olynomials and their duals can b e expressed as P l = χ ( l ) −  g l, 0 g l, 1 · · · g l,l − 1  ( g [ l ] ) − 1 χ [ l ] = ¯ S l,l  0 0 · · · 0 1  ( g [ l +1] ) − 1 χ [ l +1] = 1 det g [ l ] det        g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 1 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 x . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 x l − 1 g l, 0 g l, 1 · · · g l,l − 1 x l        , l ≥ 1 , and similar expressions for the dual polynomials. W e are now ready to get the recursion relations for orthogonal p olynomials: • First, we notice that the moment matrix g is a Hank el matrix, g i +1 ,j = g i,j +1 , which in terms of the shift matrix Λ :=   0 1 0 0 ... 0 0 1 0 ... 0 0 0 1 . . . . . . . . . . . . . . . . . .   can b e written as Λ g = g Λ > . 1 • Second, we observe the eigen-v alue prop erty Λ χ ( x ) = xχ ( x ). 1 F rom this symmetry property it follows, b y con tradiction, that the moment matrix is not in vertible; i.e. the assumption of the existence of g − 1 leads to g − 1 Λ = Λ > g − 1 , and therefore the ﬁrst row and column of g − 1 are identically zero, so that g − 1 is not inv ertible. 5 • Third, we in tro duce the LU factorization to get Λ S − 1 ¯ S = S − 1 ¯ S Λ > ⇒ S Λ S − 1 = ¯ S Λ > ¯ S − 1 =: J . F rom this last relation we deduce that the matrix J =    a 0 1 0 0 ... b 1 a 1 1 0 ... 0 b 2 a 2 1 . . . . . . . . . . . . . . . . . .    is a tridiagonal matrix, i.e. a Jacobi matrix. • Finally , w e notice that the p olynomial strings are eigen vectors of the Jacobi matrix: J P ( x ) = S Λ S − 1 S χ ( x ) = S Λ χ ( x ) = S xχ ( x ) = xP ( x ); i.e., the recursion relations xP k ( x ) = P k +1 ( x ) + a k P k ( x ) + b k P k − 1 ( x ), k > 0, hold. W e now consider the Aitken–Berg–Collar (ABC) theorem (here we follow the nomenclature used [35]) for orthogonal p olynomials. First we introduce the Christoﬀel–Darb oux kernel and therefore we consider H [ l ] = R { 0 . . . , x l − 1 } , H = n X 0 ≤ l ∞ c l x l , c l ∈ R o , ( H [ l ] ) ⊥ = n X l ≤ k ∞ c l P l ( x ) , c l ∈ R o and the resolution of the iden tity H = H [ l ] ⊕ ( H [ l ] ) ⊥ , with the corresponding orthogonal pro jector π ( l ) suc h that k er π ( l ) = ( H [ l ] ) ⊥ and Ran π ( l ) = H [ l ] . Then, the Christoﬀel–Darboux is deﬁned as K [ l ] ( x, y ) := l − 1 X k =0 P k ( y ) ¯ P k ( x ) = l − 1 X k =0 h − 1 k P k ( y ) P k ( x ) , whic h, according to the bi-orthogonalit y prop ert y , gives the following in tegral represen tation of the pro jection operator ( π ( l ) f )( y ) = Z K [ l ] ( x, y ) f ( x )d µ ( x ) , ∀ f ∈ H , An y semi-inﬁnite vector v can b e written in blo c k form as follows v = v [ l ] v [ ≥ l ] ! v [ l ] is the ﬁnite v ector formed with the ﬁrst l co eﬃcien ts of v and v [ ≥ l ] the semi-inﬁnite v ector formed with the remaining co eﬃcien ts. This decomp osition induces the following blo c k structure for any semi-inﬁnite matrix. g = g [ l ] g [ l, ≥ l ] g [ ≥ l,l ] g [ ≥ l ] ! . Giv en a factorizable moment matrix g w e hav e g [ l ] = ( S [ l ] ) − 1 ¯ S [ l ] , ( S − 1 ) [ l ] = ( S [ l ] ) − 1 , ( ¯ S − 1 ) [ ≥ l ] = ( ¯ S [ ≥ l ] ) − 1 . The Christoﬀel–Darb oux kernel is related to the moment matrix in the following wa y K [ l ] ( x, y ) = ( χ [ l ] ( x )) > ( g [ l ] ) − 1 χ [ l ] ( y ) whic h is a consequence of the follo wing iden tities K [ l ] ( x, y ) = (Π [ l ] ¯ P ( x )) > (Π [ l ] P ( y )) = χ > ( x ) ¯ S − 1 Π [ l ] S χ ( y ) = χ > ( x )(Π [ l ] ¯ S − 1 Π [ l ] )(Π [ l ] S Π [ l ] ) χ ( y ) = ( χ [ l ] ( x )) > ( g [ l ] ) − 1 χ [ l ] ( y ) . The relations ( g [ l ] ) − 1 Λ [ l ] − (Λ [ l ] ) > ( g [ l ] ) − 1 = ( g [ l ] ) − 1  g [ l, ≥ l ] (Λ [ l, ≥ l ] ) > − Λ [ l, ≥ l ] g [ ≥ l,l ]  ( g [ l ] ) − 1 follo w from the blo c k equation Λ [ l ] g [ l ] + Λ [ l, ≥ l ] g [ ≥ l,l ] = g [ l ] (Λ [ l ] ) > + g [ l, ≥ l ] (Λ [ l, ≥ l ] ) > . 6 W e also hav e Λ [ l ] χ [ l ] ( x ) = xχ [ l ] ( x ) − Λ [ l, ≥ l ] χ [ ≥ l ] ( x ) , Λ [ l, ≥ l ] = e l − 1 e > 0 , where { e i } ∞ i =0 is the canonical linear basis of H . With all these at hand we deduce ( χ [ l ] ( x )) >  ( g [ l ] ) − 1 Λ [ l ] − (Λ [ l ] ) > ( g [ l ] ) − 1  χ [ l ] ( y ) = ( χ [ l ] ( x )) > ( g [ l ] ) − 1  g [ l, ≥ l ] (Λ [ l, ≥ l ] ) > − Λ [ l, ≥ l ] g [ ≥ l,l ]  ( g [ l ] ) − 1 χ [ l ] ( y ) so that, ( x − y ) K [ l ] ( x, y ) =  ( χ [ ≥ l ] ( x )) > − ( χ [ l ] ( x )) > ( g [ l ] ) − 1 g [ l, ≥ l ]  e 0 e > l − 1 ( g [ l ] ) − 1 χ [ l ] ( y ) − ( χ [ l ] ( x )) > ( g [ l ] ) − 1 e l − 1 e > 0  χ [ ≥ l ] ( y ) − g [ ≥ l,l ] ( g [ l ] ) − 1 χ [ l ] ( y )  . That using the determinantal expressions for the p olynomials presented b efore leads to the Christoﬀel–Darb oux formula ( x − y ) K [ l ] ( x, y ) = h − 1 l − 1 ( P l ( x ) P l − 1 ( y ) − P l − 1 ( x ) P l ( y )) . 2 Multiple orthogonal p olynomials and Gauss–Borel factorization 2.1 The momen t matrix In this section we deﬁne the moment matrix in terms of a measure µ ∈ M (∆) and tw o systems of weigh ts ~ w 1 and ~ w 2 on ∆ ⊂ R , as well as corresp onding compositions (the order matters) ~ n 1 = ( n 1 , 1 , . . . , n 1 ,p 1 ) ∈ N p 1 and ~ n 2 = ( n 2 , 1 , . . . , n 2 ,p 2 ) ∈ N p 2 [38]. W e will consider multi-indices of p ositiv e integers ~ n = ( n 1 , . . . , n p ), where p ∈ N and n a ∈ Z + , a = 1 , . . . , p and deﬁne | ~ n | := n 1 + · · · + n p . F ollowing [9, 38] we observe that an y i ∈ Z + := { 0 , 1 , 2 , . . . } determines unique non-negative integers q ( i ) , a ( i ) , r ( i ), suc h that the comp osition i = q ( i ) | ~ n | + n 1 + · · · + n a ( i ) − 1 + r ( i ) , 0 ≤ r ( i ) < n a ( i ) , (5) holds. Hence, given i there is a unique k ( i ) with k ( i ) = q ( i ) n a ( i ) + r ( i ) , 0 ≤ r ( i ) < n a ( i ) . (6) Let us introduce the function integer part function [ · ] : R + → Z + , [ x ] = max { y ∈ Z + , y ≤ x } . Com bining (5) and (6) w e can obtain a formula which expresses explicitly the dep endence b et ween the quantities i , k and a i =  k n a  ( | ~ n | − n a ) + n 1 + · · · + n a − 1 + k . (7) Let R ∞ denote the v ector space of all sequences with elemen ts in R . An elemen t λ ∈ R ∞ ma y b e in terpreted as a column semi-inﬁnity vector as follows λ = ( λ (0) , λ (1) , . . . ) > , λ ( j ) ∈ R , j = 0 , 1 , . . . . W e consider the set { e j } j ≥ 0 ⊂ R ∞ with e j = ( j z }| { 0 , 0 , . . . , 0 , 1 , 0 , 0 , . . . ) > . Here ( · ) > denotes the transposition function on v ectors and matrices. Analogously , we denote b y ( R p ) ∞ the set of all sequences of v ectors with p comp onen ts and observe that each sequence whic h b elongs to ( R p ) ∞ can also b e understo od as semi-inﬁnit y column vector: giv en the v ector sequence ( ~ v 0 , ~ v 1 , . . . ) with ~ v j = ( v j, 1 , . . . , v j,p ) > w e ha ve the corresp onding sequence in R ∞ giv en by ( v 0 , 1 . . . , v 0 ,p , v 1 , 1 , . . . , v 1 ,p , . . . ); i.e., R ∞ ∼ = ( R p ) ∞ . Therefore, w e consider also the set { e a ( k ) } a =1 ,...,p k =0 , 1 ,... ⊂ ( R p ) ∞ where for each pair ( a, k ) ∈ { 1 , . . . , p } × Z + e a ( k ) = e i ( k,a ) and the function i ( a, k ) ∈ Z + satisﬁes the equality (7). No w, we are ready to introduce the monomial strings χ a := ∞ X k =0 e a ( k ) z k , χ ( l ) a = ( z k ( l ) , a = a ( l ) , 0 , a 6 = a ( l ) , χ ∗ a : = z − 1 χ a ( z − 1 ) . (8) 7 These v ectors ma y be understo od as sequences of monomials according to the comp osition ~ n in tro duced previously . W e also deﬁne the following weigh ted monomial string ξ := p X a =1 χ a w a , ξ ( l ) = w a ( l ) z k ( l ) , (9) whic h is a sequence of weigh ted monomials for eac h given comp osition ~ n . Sometimes, when we what to stress the dep endence in the composition w e write χ ~ n,a , χ ~ n and ξ ~ n . Given the weigh ted monomials ξ ~ n 1 and ξ ~ n 2 , associated to the comp ositions ~ n 1 and ~ n 2 , we introduce the moment matrix in the following manner Deﬁnition 1. The moment matrix is given by g ~ n 1 ,~ n 2 := Z ξ ~ n 1 ( x ) ξ ~ n 2 ( x ) > d µ ( x ) . (10) In terms of the canonical basis { E i,j } of the linear space of semi-inﬁnite matrices and for each pair ( i, j ) ∈ Z 2 + w e consider the binary p ermutations or transpositions π i,j = E i,j + E j,i . Observ e that π 2 i,j = I and therefore π − 1 i,j = π i,j . Giv en tw o transp ositions π i,j and π k,l the p erm utation endomorphism corresp onding to its pro duct is well deﬁned π i,j π k,l = π k,l π i,j . T aking a sequence of pairs I = { ( i s , j s ) } s ∈ Z + , i s , j s ∈ Z + , w e introduce the permutation endomorphism π I as the inﬁnite pro duct π I = Q s ∈ Z + π i s ,j s , with π I π > I = π > I π I = I . Giv en t wo comp ositions, ~ n 0 , ~ n , there exists a p erm utation π ~ n 0 ,~ n suc h that ξ ~ n 0 = π ~ n 0 ,~ n ξ ~ n through a p erm utation semi-inﬁnite matrix as just describ ed. The change in the comp ositions is mo deled as follows Prop osition 1. Given two set of weights ~ w ` = ( w `, 1 , . . . , w `,p ` ) and c omp ositions ~ n ` and ~ n 0 ` , ` = 1 , 2 , ther e exist p ermutation matric es π ~ n 0 ` ,~ n ` such that g ~ n 0 1 ,~ n 0 2 = π ~ n 0 1 ,~ n 1 g ~ n 1 ,~ n 2 π > ~ n 0 2 ,~ n 2 . (11) Pr o of. F or an y set of weigh ts ~ w = w 1 , . . . , w p and tw o comp ositions ~ n and ~ n 0 w e hav e that the corresp onding vectors of weigh ted monomials are connected, ξ ~ n 0 = π ~ n 0 ,~ n ξ ~ n trough a p erm utation semi-inﬁnite matrix; i.e, π > ~ n 0 ,~ n = π − 1 ~ n 0 ,~ n . Therefore, the announced result follo ws. F or the sake of notation simplicity and when the context is clear enough we will drop the subindex indicating the t wo comp ositions and just write g for the momen t matrix. Let us discuss in more detail the blo ck Hank el structure of the momen t matrix. F or eac h pair ( i, j ) ∈ Z 2 + there exists a unique com bination of three others pairs ( q 1 , q 2 ) ∈ Z 2 + , ( a 1 , a 2 ) ∈ { 1 , . . . , p 1 } × { 1 , . . . , p 2 } and ( r 1 , r 2 ) ∈ { 0 , . . . , n 1 ,a 1 − 1 } × { 0 , . . . , n 2 ,a 2 − 1 } , such that i = q 1 | ~ n 1 | + n 1 , 1 + . . . + n 1 ,a 1 − 1 + r 1 and j = q 2 | ~ n 2 | + n 2 , 1 + . . . + n 2 ,a 2 − 1 + r 2 . Hence taking k ` = q ` n `,a ` + r ` , ` = 1 , 2 , the coeﬃcients g i,j ∈ R of the moment matrix g = ( g i,j ) hav e the follo wing explicit form g i,j = Z x k 1 + k 2 w 1 ,a 1 ( x ) w 2 ,a 2 ( x )d µ ( x ) . (12) Observ e that pairs ( k 1 , a 1 ) and ( k 2 , a 2 ) are univocally determined by i and j respectively . Before w e contin ue with the study of this momen t matrix it is necessary to introduce some auxiliary ob jects asso ciated with the vector space R ∞ . First, we hav e the unity matrix I = P ∞ k =0 e k e > k and the shift matrix Λ := P ∞ k =0 e k e > k +1 . W e also deﬁne the pro jections Π [ l ] := P l − 1 k =0 e k e > k , and with the help of the set { e a ( k ) } a =1 ,...,p k =0 , 1 ,... w e construct the pro jections Π a := P ∞ k =0 e a ( k ) e a ( k ) > with P p a =1 Π a = I , and P 1 := diag( I n 1 , 0 n 2 , . . . , 0 n p ) , P 2 := diag(0 n 1 , I n 2 , . . . , 0 n p ) , . . . P p := diag(0 n 1 , 0 n 2 , . . . , I n p ) , (13) where I n s is the n s × n s iden tity matrix. Finally w e in tro duce the notation x ~ n := x n 1 P 1 + · · · + x n p P p = diag( x n 1 I n 1 , . . . , x n p I n p ) : R → R | ~ n |×| ~ n | . (14) 8 F or a b etter insight of the moment matrix let us introduce the following n 1 ,a × n 2 ,b matrices m a,b ( x ) = w 1 ,a ( x ) w 2 ,b ( x )      1 x · · · x n 2 ,b − 1 x x 2 · · · x n 2 ,b . . . . . . . . . x n 1 ,a − 1 x n 1 ,a · · · x n 1 ,a + n 2 ,b − 2      , a = 1 , . . . , p 1 , b = 1 , . . . , p 2 , (15) in terms of which we build up the following | ~ n 1 | × | ~ n 2 | -matrix m :=      m 1 , 1 m 1 , 2 · · · m 1 ,p 2 m 2 , 1 m 2 , 2 · · · m 2 ,p 2 . . . . . . . . . m p 1 , 1 m p 1 , 2 · · · m p 1 ,p 2      : R → R | ~ n 1 |×| ~ n 2 | . (16) Then, the moment matrix g has the following blo c k structure g := ( G i,j ) i,j ≥ 0 ∈ R ∞×∞ , G i,j := Z x i~ n 1 m ( x ) x j ~ n 2 d µ ( x ) ∈ R | ~ n 1 |×| ~ n 2 | . (17) Fix now a num b er l ∈ N and consider the pair ( l , l + 1). There exists a unique com bination of pairs ( q 1 , q 2 ) ∈ Z 2 + , ( a 1 , a 2 ) ∈ { 1 , . . . , p 1 } × { 1 , . . . , p 2 } and ( r 1 , r 2 ) ∈ { 0 , . . . , n 1 ,a 1 − 1 } × { 0 , . . . , n 2 ,a 2 − 1 } , such that l = q 1 | ~ n 1 | + n 1 , 1 + · · · + n 1 ,a 1 − 1 + r 1 and l + 1 = q 2 | ~ n 2 | + n 2 , 1 + · · · + n 2 ,a 2 − 1 + r 2 . Giv en the comp ositions ~ n 1 and ~ n 2 w e introduce the degree m ulti-indices ~ ν 1 ∈ Z p 1 + and ~ ν 2 ∈ Z p 2 + [9] where for each ` = 1 , 2 , w e hav e ~ ν ` = ( ν `, 1 , . . . , ν `,a ` − 1 , ν `,a ` , ν `,a ` +1 , . . . , ν `,p ` ) = (( q ` + 1) n `, 1 , . . . , ( q ` + 1) n `,a ` − 1 , q ` n `,a ` + r ` , q ` n `,a ` +1 , . . . , q ` n `,p ` ) , (18) whic h satisfy k ` ( i + 1) = ν `,a ` ( i ) ( i ) , | ~ ν ` ( i ) | = i + 1 , ~ ν ` ( i + l | ~ n ` | ) = ~ ν ` ( i ) + l~ n ` , (19) and consider the l × ( l + 1) blo c k matrix Γ l from g Γ l =      g 0 , 0 g 0 , 1 · · · g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l      . (20) Let us study the homogeneous system Γ l x l +1 = 0 l , where x l +1 ∈ R l +1 and 0 l +1 is the null v ector in R l +1 . T aking into accoun t Γ l ’s structure (12), we see that such equation is exactly the expression of the orthogonality relations (3). W e can see now that for each l ∈ N the existence of a system of mixed m ultiple orthogonal p olynomials  A ( l ) 1 , . . . , A ( l ) p 1  is ensured; that is because Γ l in (20) is a l × ( l + 1) matrix, and the homogeneous matrix equation Γ l x l +1 = 0 l , whic h is satisﬁed b y the co eﬃcien ts of the polynomials corresp onds to a system of l homogeneous linear equations for l + 1 unkno wn coeﬃcients. Thus, the system alw ays has a non-trivial solution. Obviously ,  A ( l ) 1 , . . . , A ( l ) p 1  is not univ o cally determined by the matrix equation Γ l x l +1 = 0 l or equiv alen tly b y the orthogonality relations (3), b ecause its solution space has at least dimension 1. Hence, the appropriate question to consider is the uniqueness question without coun ting constan t factors, or equiv alently if the solution space has exactly dimension 1. In terms of Γ l the question b ecomes: Do es Γ l ha ve rank l ? In order to hav e a positive answer it is suﬃcient to ensure that the l × l square matrix g [ l ] :=      g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1      , l ≥ 1 , (21) is in v ertible, where g [ l ] results from Γ l after remo ving its last column. It is easy to pro ve that suc h condition is equiv alent to require that all p ossible solutions of (3) satisfy deg A p 1 = ν 1 ,p 1 − 1. Ob viously this requiremen t is ensured when the p olynomials  A ( l ) 1 , . . . , A ( l ) p 1  fulﬁll deg A j = ν 1 ,j − 1 , j = 1 , . . . , p 1 . 9 2.2 P erfect com binations and Nikishin systems W e in tro duce the concept of a p erfect combination. Deﬁnition 2. A c ombination ( µ, ~ w 1 , ~ w 2 ) of a me asur e µ ∈ M (∆) and two systems of weights ~ w 1 and ~ w 2 on ∆ ⊂ R is said to b e p erfe ct if for e ach p air of multi-indic es ( ~ ν 1 , ~ ν 2 ) , with | ~ ν 1 | = | ~ ν 2 | + 1 the ortho gonality r elations (3) imply that deg A a = ν 1 ,a − 1 , a = 1 , . . . , p 1 . F or a p erfect combination ( µ, ~ w 1 ,~ ,w 2 ) and any given l ∈ Z + the solution space of the equation Γ l x l +1 = 0 l is one- dimensional. Then, we can determine a unique system of mixed t yp e orthogonal p olynomials  A 1 , . . . , A p 1  satisfying (3) requiring for a 1 ∈ { 1 , . . . p 1 } that A a 1 monic. F ollowing [14] we sa y that we hav e a type I I normalization and denote the corresp onding system of polynomials by A (II ,a 1 ) a , j = 1 , . . . , p 1 . Alternatively , w e can proceed as follo ws, since the system of weigh ts is perfect from (3) we deduce that Z x ν 2 ,b 2 p 1 X a =1 A a ( x ) w 1 ,a ( x ) w 2 ,b 2 ( x )d µ ( x ) 6 = 0 . Then, we can determine a unique system of mixed type of multi-orthogonal p olynomials ( A (I ,b 2 ) 1 , . . . , A (I ,b 2 ) p 2 ) imp osing that Z x ν 2 ,b 2 p 1 X a =1 A (I ,b 2 ) a ( x ) w 1 ,a ( x ) w 2 ,b ( x )d µ ( x ) = 1 , whic h is a type I normalization. W e will use the notation A (II ,a 1 ) [ ~ ν 1 ; ~ ν 2 ] ,a and A (I ,b 2 ) [ ~ ν 1 ; ~ ν 2 ] ,a to denote these multiple orthogonal p olynomials with type I I and I normalizations, resp ectively . A kno wn illustration of p erfect com binations ( µ, ~ w 1 , ~ w 2 ) can b e constructed with an arbitrary p ositive ﬁnite Borel measure µ and systems of weigh ts formed with exp onen tials: (e γ 1 x , . . . , e γ p x ) , γ i 6 = γ j , i 6 = j, i, j = 1 , . . . , p, (22) or by binomial functions ((1 − z ) α 1 , . . . , (1 − z ) α p ) , α i − α j 6∈ Z , i 6 = j, i, j = 1 , . . . , p. (23) or com bining b oth classes, see [31]. Recen tly a wide class of systems of w eights where pro ven to be p erfect [20]; these systems of functions, now called Nikishin systems, were introduced by E.M. Nikishin [31] and initially named MT-systems (after Marko v and Tcheb ycheﬀ ). Giv en a closed interv al ∆ let ◦ ∆ b e the interior set of ∆ . Let us take tw o interv als ∆ α and ∆ β whose in terior sets are disjoint, i.e. ◦ ∆ α ∩ ◦ ∆ β = ∅ . Set tw o measures µ α ∈ M (∆ α ) and µ β ∈ M (∆ β ) such that the measure h µ α , µ β i with the following diﬀerential form d h µ α , µ β i ( x ) = Z d µ β ( t ) x − t d µ α ( x ) = b µ β ( x )d µ α ( x ) , is a ﬁnite measure, that implies that h µ α , µ β i ∈ M (∆ α ) . The function b µ β denotes the Cauc hy tr ansform corresp onding to µ β . Let us consider then a system of p in terv als ∆ 1 , . . . , ∆ p suc h that ◦ ∆ j ∩ ◦ ∆ j +1 = ∅ , j ∈ { 1 , . . . , p − 1 } . T ake p measures µ j ∈ M (∆ j ) , which for each j = 1 , . . . , p − 1 , the measure h µ j , σ j +1 i b elongs to M (∆ j ). So the system of measures ( ξ 0 , . . . , ξ p ) where ζ 1 = µ 1 , ζ 2 = h µ 1 , µ 2 i , ζ 3 = h µ 1 , h µ 2 , µ 3 ii = h µ 1 , µ 2 , µ 3 i , . . . , ζ p = h µ 1 , . . . , µ p i , is the Nikishin system of measures generated b y the system ( σ 1 , . . . , σ m ). So w e denote ( ζ 1 , . . . , ζ p ) = N ( σ 1 , . . . , σ p ) . Actually , in [20] the authors shown perfectness for com binations of Nikishin systems where in terv als ∆ 1 , . . . , ∆ p are b ounded and for each j ∈ { 1 , . . . , p − 1 } the interv als ∆ j and ∆ j +1 are disjoint. The same authors hav e communicated to us that they w ere able to prov e a generalization of this result to un b ounded in terv als suc h that ∆ j ∩ ∆ j +1 6 = ∅ . Consequen tly , in what follo ws w e assume such generalization. 10 As we hav e seen, general Nikishin systems hav e an in tricate structure; therefore, in order to mak e easy the reader w e focus on a “simple” class of Nikishin systems which w e call M-Nikishin systems. Set the in terv al ∆ 1 = [0 , 1] and let M 0 (∆ 1 ) ⊂ M (∆ 1 ) denote the set of measures in M (∆ 1 ) such that if σ ∈ M 0 (∆ 1 ) then, the function e σ ( x ) := Z ∆ 1 d σ ( t ) 1 − tx satisﬁes lim x → 1 x ∈ ◦ ∆ 1     Z ∆ 1 d σ ( t ) 1 − tx     = lim x → 1 x ∈ ◦ ∆ 1 Z ∆ 1 | d σ ( t ) | 1 − tx < + ∞ , where ◦ ∆ 1 = (0 , 1) . (24) The constrain t (24) guarantees that the function e σ is a weigh t in compact in terv als in ( −∞ , 1]. As (1 − tx ) do es not v anish for ( t, x ) ∈ ∆ 1 × ( C \ [1 , + ∞ )) we deduce that 1 / (1 − tx ) is a contin uous function in x for t ∈ ∆ 1 . Therefore, we conclude that e σ is a holomorphic function on C \ (1 , + ∞ ), having a contin uation as contin uous function in 1 . T aking in to account that e σ do es not v anish in C \ (1 , + ∞ ) and that it tak es real v alues on R \ (1 , + ∞ ) = ( −∞ , 1], w e deduce that it is a contin uous weigh t on ( −∞ , 1]. Observe that e σ ( x ) = Z ∆ 1 dσ ( t ) 1 − tx = Z [1 , + ∞ ) ζ dσ (1 /ζ ) x − ζ = Z [1 , + ∞ ) d µ ( ζ ) x − ζ , (25) is the Cauch y transform of another measure µ ∈ M ([1 , + ∞ )) , such that | b µ (1) | = | e σ (1) | < + ∞ . Giv en t wo measures σ α ∈ M 0 (∆ 1 ) , σ β ∈ M 0 (∆ 1 ) we deﬁne a third one as follows (using the diﬀerential notation) d[ σ α , σ β ]( x ) = e σ β ( x )d σ α ( x ) , e σ β ( x ) = Z ∆ 1 d σ β ( ζ ) 1 − xζ . As e σ β is a con tinuous weigh t on ∆ 1 w e conclude that [ σ α , σ β ] ∈ M 0 (∆ 1 ). If we take a system of measures ( σ 1 , . . . , σ p ) suc h that σ j ∈ M 0 (∆ 1 ) , j = 1 , . . . , p , we say that ( s 1 , . . . , s p ) = MN ( σ 1 , . . . , σ p ), where s 1 = σ 1 , s 2 = [ σ 1 , σ 2 ] , s 3 = [ σ 1 , [ σ 2 , σ 3 ]] = [ σ 1 , σ 2 , σ 3 ] , . . . s p = [ σ 1 , σ 2 , . . . , σ p ] (26) is the M-Nikishin system of measures generated by ( σ 1 , . . . , σ p ), with corresp onding M-Nikishin system of functions giv en by ~ w = ( w 1 , . . . , w p ) = ( e s 1 , . . . , e s p ) = M b N ( σ 1 , . . . , σ p ). Notice that s i ∈ M 0 (∆ 1 ) whic h implies that for each arbitrary compact subinterv al of ( −∞ , 1] the system of functions ~ w conforms a system of contin uous weigh ts. M-Nikihsin systems are included in the class of Nikishin systems. T aking into account the iden tity (25) we see that the M-Nikishin system deﬁned in (26) can b e written as a classical Nikishin system. Let us take a system ( µ 1 , . . . , µ p ) where µ 1 = σ 1 , d µ 2 ( x ) = x d σ 2 (1 /x ) , µ 3 = σ 3 , . . . , µ 2[ p/ 2] − 1 = σ 2[ p/ 2] − 1 , d µ 2[ p/ 2] ( x ) = xσ 2[ p/ 2] (1 /x ) , and if p is o dd µ p = σ p . Notice then s 1 = ζ 1 = σ 1 , s 2 = ζ 2 = h µ 1 , µ 2 i , . . . s p = ζ p = h µ 1 , µ 2 , . . . , µ p i . Hence ( s 1 , . . . , s p ) = MN ( σ 1 , . . . , σ p ) = N ( µ 1 , . . . , µ p ) = ( ζ 1 , . . . , ζ p ) . Fixing tw o M-Nikishin systems of functions ~ w ` ( x ) = ( e s `, 1 ( x ) , . . . e s `,p ( x )) whose elemen ts are w eigh ts on ∆ 0 = [ − 1 , 1], and a measure µ ∈ M (∆ 0 ) we ha ve at our disp osal the perfect com bination ( µ, ~ w 1 , ~ w 2 ). W e can also obtain a p erfect combination ( µ, ~ w 1 , ~ w 2 ) c ho osing ~ w 1 and ~ w 2 b et w een tw o diﬀerent of the classes mentioned in (22) and (23) (not necessarily the same). Prop osition 2. The T aylor series at ζ = 0 c orr esp onding to the functions e s j ( ζ ) and f j ( ζ ) := log e s j ( ζ ) c onver ge uniformly to e s j and f j r esp e ctively on ∆ 1 , i.e. e s j ( x ) = ∞ X i =0 λ i,j x i = e P ∞ i =0 t i,j x i , x ∈ ∆ 0 , j = 1 , . . . , p. (27) wher e λ i,j and t i,j ar e c onstants. Pr o of. F or each j ∈ { 1 , . . . , p } , e s j is a holomorphic function on the open unitary disc centered on the origin. That implies that e s j ( x ) = ∞ X i =0 λ i,j x i = e P ∞ i =0 t i,j x i , x ∈ {| ζ | < 1 } , j = 1 , . . . , p. Notice that      ∞ X i =0 λ i,j      = lim x → 1 x ∈ [0 , 1)      ∞ X i =0 λ i,j x i      = lim x → 1 x ∈ ◦ ∆ 1     Z d s j ( t ) 1 − xt     < + ∞ , So the ﬁrst equality in (27) is pro v ed. The second one comes immediately from the fact that the functions e s j do not v anish on ∆ 0 . That implies that P ∞ i =0 t i,j x i are also bounded and therefore con tinuous. Hence we can pro ceed analogously as in the ﬁrst equality . 11 2.2.1 The in verse problem Giv en the series w j ( x ) = ∞ X i =0 λ i,j x i = e P ∞ i =0 t i,j x i , x ∈ ∆ 0 , j = 1 , . . . , p, (28) w e consider the problem of ﬁnding conditions ov er { λ i,j } suc h that the set of series { w j } p j =1 form a M-Nikishin system of functions. The reader should notice that λ i,j = S i ( t i, 0 , t i, 1 , . . . , t i,j ) where S i is the i -th elementary Sc hur p olyno- mial. Elementary Sch ur p olynomials S j ( t 1 , . . . , t j ) are deﬁned b y the following generating relation exp( P ∞ j =1 t j z j ) = P ∞ j =0 S j ( t 1 , t 2 , . . . , t j ) z j , and therefore S j = P j p =1 P j 1 + ··· + j p = j t j 1 · · · t j p . Given a partition ~ n = ( n 1 , . . . , n r ) ∈ Z r + w e ha ve the Sch ur function s ~ n ( t ) = det( S n i − i + j ( t )) 1 ≤ i,j ≤ r . F or more on the relation of these Sch ur functions and those in [26], see [32]. In order to state suﬃcient conditions in this direction w e need some preliminary deﬁnitions and results. Deﬁnition 3. Given a se quenc e C = { c i } ∞ i =0 ⊂ R its Hausdorﬀ moment pr oblem c onsists in ﬁnding a me asur e σ ∈ M (∆) such that c i = Z ζ i d σ ( ζ ) , i ∈ Z + . Mor e over, if we further imp ose the c onstr aint σ ∈ M 0 (∆) we say that we have a r estricte d Hausdorﬀ moment pr oblem. Here w e hav e made a v ariation in the classical deﬁnition of a Hausdorﬀ problem, where the solutions are p ositiv e measures. In our Hausdorﬀ problem we lo ok for measures in a wider class where they do not change their sign. Ob viously , since M 0 (∆) ⊂ M (∆) each solution of a restricted Hausdorﬀ problem is also a solution of a Hausdorﬀ problem. In the pages 8 and 9 in [34] J. A. Shohat and J. D. T amarkin study Hausdorﬀ problems and giv e a suﬃcient and necessary condition ov er the sequences to hav e solution. Using this result we deduce the following Lemma. Lemma 1. The Hausdorﬀ moment pr oblem for a se quenc e C = { c i } ∞ i =0 ⊂ R has a solution if and only if n X i =0  n i  ( − 1) i c i + k ≥ 0 ∀ ( n, k ) ∈ Z 2 + or n X i =0  n i  ( − 1) i c i + k ≤ 0 ∀ ( n, k ) ∈ Z 2 + . (29) When (29) holds a ne c essary and suﬃcient c ondition that ensur es solution for the r estricte d Hausdorﬀ moment pr oblem of C is    ∞ X i =0 c i    < + ∞ . (30) Pr o of. Theorem 1.5 in [34] states that the ﬁrst set of inequalities in (29) is a necessary and suﬃcient condition to ha ve a p ositiv e measure σ solving the classical Hausdorﬀ problem. F ollowing their proof it is not hard to conclude that adding the second set of inequalities leads to a solution in M (∆). Let us take a measure σ ∈ M (∆) and observe that R d σ ( t ) 1 − xt is a holomorphic function on ¯ C \ [1 , + ∞ ), then if C is its moment sequence we deduce Z ∆ d σ ( t ) 1 − xt = ∞ X i =0 c i x i , x ∈ {| ζ | < 1 } . Th us, since all the c i ’s hav e the same sign, by Leb esgue’s dominated conv ergence Theorem we ha ve lim x → 1 x ∈ [0 , 1)    Z ∆ d σ ( t ) 1 − xt    = lim x → 1 x ∈ [0 , 1)    ∞ X i =0 x i c i    =    ∞ X i =0 c i    . Th us σ ∈ M 0 (∆) if and only if (30) takes place. Giv en the series w j ( x ) = ∞ X i =0 λ i,j, 1 x i , x ∈ ∆ 1 , j = 1 , . . . , p, (31) 12 w e introduce a set of semi-inﬁnite matrices Θ k and semi-inﬁnite v ectors θ j,k , j = k , . . . , p, k = 1 , . . . , p in the following recursiv e wa y . First, we deﬁne Θ 1 :=      λ 0 , 1 , 1 λ 1 , 1 , 1 λ 2 , 1 , 1 · · · λ 1 , 1 , 1 λ 2 , 1 , 1 λ 3 , 1 , 1 · · · λ 2 , 1 , 1 λ 3 , 1 , 1 λ 4 , 1 , 1 · · · . . . . . . . . . . . .      , θ j, 1 :=      λ 0 ,j, 1 λ 1 ,j, 1 λ 2 ,j, 1 . . .      , j = 1 , . . . , p. Then, w e seek solutions θ j, 2 :=    λ 0 ,j, 2 λ 1 ,j, 2 λ 2 ,j, 2 . . .    of Θ 1 θ j, 2 = θ j, 1 , for j = 2 , . . . , p , and if these solutions exist w e deﬁne Θ 2 :=    λ 0 , 2 , 2 λ 1 , 2 , 2 λ 2 , 2 , 2 ··· λ 1 , 2 , 2 λ 2 , 2 , 2 λ 3 , 2 , 2 ··· λ 2 , 2 , 2 λ 3 , 2 , 2 λ 4 , 2 , 2 ··· . . . . . . . . . . . .    . Then, w e lo ok for θ j, 3 =    λ 0 ,j, 3 λ 1 ,j, 3 λ 2 ,j, 3 . . .    whic h solves Θ 2 θ j, 3 = θ j, 2 , for j = 3 , . . . , p , and when such solutions exist we in tro duce Θ 3 =    λ 0 , 3 , 3 λ 1 , 3 , 3 λ 2 , 3 , 3 ··· λ 1 , 3 , 3 λ 2 , 3 , 3 λ 3 , 3 , 3 ··· λ 2 , 3 , 3 λ 3 , 3 , 3 λ 4 , 3 , 3 ··· . . . . . . . . .    . In this wa y we get for k ∈ { 1 , . . . , p } the matrices Θ k and vectors θ j,k , j = k , . . . , p , linked b y Θ k θ j,k +1 = θ j,k with expressions Θ k +1 =      λ 0 ,k +1 ,k +1 λ 1 ,k +1 ,k +1 λ 2 ,k +1 ,k +1 · · · λ 1 ,k +1 ,k +1 λ 2 ,k +1 ,k +1 λ 3 ,k +1 ,k +1 · · · λ 2 ,k +1 ,k +1 λ 3 ,k +1 ,k +1 λ 4 ,k +1 ,k +1 · · · . . . . . . . . . . . .      , θ j,k +1 =      λ 0 ,j,k +1 λ 1 ,j,k +1 λ 2 ,j,k +1 . . .      . Here we understand Θ k θ j,k +1 = θ j,k as ∞ X i =0 λ l + i,k,k λ i,j,k +1 = λ l,j,k , l ∈ Z + . W e no w consider the sequences C k,k := { λ i,k,k } ∞ i =0 , C j,k := { λ i,j,k } ∞ i =0 , j = k , . . . , p, k = 1 , . . . , p. (32) Later, we will prov e that none of the semi-inﬁnite Hank el matrices Θ k , k = 1 , . . . , p , are in vertible. Hence such inﬁnite linear systems are either undetermined or incompatible. In this last case we say that the systems of sequences ( C k,k , . . . , C p,k ), k = 1 , . . . , p , do not exist. First we need the following preliminary Lemma 2. The series w ( x ) = ∞ X i =0 λ i x i , x ∈ ∆ 0 , c onver ges uniformly on ∆ 0 to a function e σ ( x ) = R d σ ( t ) / (1 − tx ) c orr esp onding to a me asur e σ ∈ M 0 (∆ 1 ) on ∆ 0 if and only if the r estricte d Hausdorﬀ moment pr oblem c orr esp onding to the se quenc e { λ i : i ∈ Z + } has a solution. Pr o of. Let us assume that the restricted Hausdorﬀ momen t problem of a sequence { λ i : i ∈ Z + } has a solution. That means that there exists a measure σ ∈ M 0 (∆) such that λ i = Z ∆ 1 t i d σ ( t ) , i ∈ Z + , lim x → 1 x ∈ [0 , 1)     Z ∆ 1 d σ ( t ) 1 − tx     =      ∞ X i =0 λ i      < + ∞ . Since | λ i x i | ≤ | λ i | , | x | ≤ 1 and P ∞ i =0 | λ i | < + ∞ , by W eirestrass’ Theorem P ∞ i =0 λ i x i con verges uniformly on ∆ 0 . This pro ves the if implication in the Lemma 2. On the other hand lim x → 1 x ∈ [0 , 1)     Z ∆ d σ ( t ) 1 − tx     = lim x → 1 x ∈ [0 , 1)      ∞ X i =0 λ i x i      =      ∞ X i =0 λ i      b ecause   P ∞ i =0 λ i x i   m ust be contin uous on ∆ 0 . λ i coincides with the i -th moment corresp onding to the measure σ whic h completes the pro of. 13 Theorem 1. The system of weights { w 1 ,j } p j =1 , as in (31) , c onver ges uniformly in ∆ 0 to an M-Nikishin system of functions { b s j } p j =1 if and only if for e ach k = 1 , . . . , p − 1 , ther e exists a system of se quenc es ( C k,k , . . . , C p,k ) as in (32) , such that their r estricte d Hausdorﬀ moment pr oblems have solutions. Pr o of. The pro of of this Theorem go es as follows. F rom Lemma 2 we ha ve that for each j = 1 , . . . , p, w j, 1 ( x ) = ∞ X i =0 λ i,j, 1 x i , x ∈ ∆ 0 , con verges in ∆ 0 to a function e s j ( x ) = R d s j ( t ) / (1 − tx ) if and only if the restricted Hausdorﬀ moment problem corre- sp onding to { λ i,j, 1 : i ∈ Z + } has a solution. W e assume that w j, 1 con verges uniformly on ∆ 0 to the function e s j corre- sp onding to the s j ∈ M (∆ 1 ) . In order to prov e the necessity in Theorem’s statement we supp ose that ( s 1 , . . . , s p ) = MN ( σ 1 , . . . , σ p ) is an M-Nikishin system of measures as it was deﬁned in the § 2.2. Fixed k ∈ { 1 , . . . , p } we deﬁne another M-Nikishin system ( s k,k , . . . , s k,p ) = MN ( σ k , . . . , σ p ) . Let us observe that ( s 1 , 1 , . . . , s 1 ,p ) = ( s 1 , . . . , s p ) . By construction for each k ∈ { 1 , . . . , p } , w e ha ve that d s k,j = e s k +1 ,j d s k,k , j = k , . . . , p. When j = k w e understand e s k +1 ,k ≡ 1 . Fixed j ∈ { k + 1 , . . . , p } , e s k +1 ,j is a holomorphic function on ¯ C \ ( −∞ , 1); hence, its T aylor’s series w j,k +1 ( t ) = ∞ X i =0 λ i,j,k +1 t i , t ∈ ∆ 1 ⊂ ∆ 0 con verges uniformly to e s k +1 ,j on ∆ 1 . Then, for eac h x ∈ ∆ 1 e s k,j ( x ) = ∞ X l =0 λ l,j,k x l = Z R e s k +1 ,j ( t ) d s k,k ( t ) 1 − tx = Z R ∞ X i =0 λ i,j,k +1 t i d s k,k ( ζ ) 1 − tx = = ∞ X l =0 ∞ X i =0 λ i,j,k +1 Z R t i + l d s k,k ( t ) = ∞ X l =0 ∞ X i =0 λ i,j,k +1 λ l + i,k,k x l , whic h prov es one implication of the equiv alence. The other implication comes immediately from Lemma 2. W e remark from the statements of Lemma 1 that the conditions in Theorem 1 are equiv alent to the inequalities in (29). Hence, by con tinuit y criteria, such conditions are stable under p erturbations of the co eﬃcien ts λ i, 1 , 1 , i ∈ Z + . W e will come to this later in § 3, when w e consider deformations of the w eights leading to the multicomponent 2D T o da ﬂo ws in the precise form discussed in this section. 2.3 The Gauss–Borel factorization and multiple orthogonal p olynomials Giv en a p erfect combination ( µ, ~ w 1 , ~ w 2 ) we consider [2] Deﬁnition 4. The Gauss–Bor el factorization (also known as LU factorization) of a semi-inﬁnite moment matrix g , determine d by ( µ, ~ w 1 , ~ w 2 ) , is the pr oblem of ﬁnding the solution of g = S − 1 ¯ S , S =      1 0 0 · · · S 1 , 0 1 0 · · · S 2 , 0 S 2 , 1 1 · · · . . . . . . . . . . . .      , ¯ S − 1 =      ¯ S 0 0 , 0 ¯ S 0 0 , 1 ¯ S 0 0 , 2 · · · 0 ¯ S 0 1 , 1 ¯ S 0 1 , 2 · · · 0 0 ¯ S 0 2 , 2 · · · . . . . . . . . . . . .      , S i,j , ¯ S 0 i,j ∈ R . (33) In terms of these matric es we c onstruct the p olynomials A ( l ) a := X 0 i S l,i x k 1 ( i ) , (34) wher e the sum P 0 is taken for a ﬁxe d a = 1 , . . . , p 1 over those i such that a = a 1 ( i ) and i ≤ l . We also c onstruct the dual p olynomials ¯ A ( l ) b := X 0 j x k 2 ( j ) ¯ S 0 j,l , (35) wher e the sum P 0 is taken for a given b over those j such that b = a 2 ( j ) and j ≤ l . 14 This factorization makes sense whenever all the principal minors of g do not v anish, i.e., if det g [ l ] 6 = 0 l = 1 , 2 , . . . , and in our case it is true because ( µ, ~ w 1 , ~ w 2 ) is a p erfect combination. It can b e shown that the following sets G − := n S =   1 0 0 ··· S 1 , 0 1 0 ··· S 2 , 0 S 2 , 1 1 . . . . . . . . . . . .   , S i,j ∈ R o , G + := n ¯ S =    ¯ S 0 , 0 ¯ S 0 , 1 ¯ S 0 , 2 ··· 0 ¯ S 1 , 1 ¯ S 1 , 2 ··· 0 0 ¯ S 2 , 2 ··· . . . . . . . . . . . .    , ¯ S i,j ∈ R , ¯ S i,i 6 = 0 o are groups. Indeed, the multiplication of tw o arbitrary semi-inﬁnite matrices is, in general, not well deﬁned as it in volv es, for eac h coeﬃcient of the pro duct, a series; how ever if the tw o matrices lie on G − , the mentioned series collapses in to a ﬁnite sum, and the same holds for G + . Moreo ver, the in v erse of a matrix in S ∈ G − can be found to b e in G − in a recursiv e wa y: ﬁrst we express S = I + P i> 0 S i (Λ > ) i with S i = diag( S i (0) , S i (1) , . . . ) a diagonal matrix, then we assume S − 1 = I + P i> 0 ˜ S i (Λ > ) i to hav e the same form, and ﬁnally we ﬁnd that the diagonal matrix unkno wn coeﬃcients ˜ S i are expressed in terms of S 0 , . . . , S i in a unique wa y; the same holds in G + . Giv en, t wo elemen ts S ∈ G − and ¯ S ∈ G + the co eﬃcients of the pro duct S ¯ S are ﬁnite sums. Ho w ever, this is not the case for ¯ S S , where the co eﬃcien ts are series. Therefore, given an LU factorizable elemen t g = S − 1 ¯ S w e can not ensure that g has an in verse, observ e that in spite of the existence of S and ¯ S − 1 , the existence of ¯ S − 1 S = g − 1 is not ensured as this pro duct in volv es the ev aluation of series instead of ﬁnite sums. With the use of the co eﬃcients of the matrices S and ¯ S we construct multiple orthogonal polynomials of mixed t yp e with normalizations of type I and II Prop osition 3. We have the fol lowing identiﬁc ations A ( l ) a = A (II ,a 1 ( l )) [ ~ ν 1 ( l ); ~ ν 2 ( l − 1)] ,a , ¯ A ( l ) b = A (I ,a 1 ( l )) [ ~ ν 2 ( l ); ~ ν 1 ( l − 1)] ,b , in terms of multiple ortho gonal p olynomials of mixe d typ e with two normalizations I and II , r esp e ctively. Pr o of. F rom the LU factorization we deduce l X i =0 S l,i g i,j = 0 , j = 0 , 1 , . . . , l − 1 , S ii := 1 . (36) With the aid of (18) and (34) w e express (36) as follows Z  p 1 X a =1 A ( l ) a ( x ) w 1 ,a ( x )  w 2 ,b ( x ) x k d µ ( x ) = 0 , deg A ( l ) a ≤ ν 1 ,a ( l ) − 1 , (37) 0 ≤ k ≤ ν 2 ,b ( l − 1) − 1 . W e recognize these equations as those deﬁning a set of m ultiple orthogonal p olynomials of mixed type as discussed in [14]. This fact leads to A [ ~ ν 1 ; ~ ν 2 ] ,a := A ( l ) a where ~ ν 1 = ~ ν 1 ( l ) and ~ ν 2 = ~ ν 2 ( l − 1). Observ e that for a given l eac h p olynomial A [ ~ ν 1 ; ~ ν 2 ] ,a has at m uch ν 1 ,a ( l ) coeﬃcients, and therefore we hav e | ~ ν 1 ( l ) | = l + 1 unkno wns, while we hav e | ~ ν 2 ( l − 1) | = l equations. Moreo ver, from the normalization condition S ii = 1 we get that the p olynomial A [ ~ ν 1 ; ~ ν 2 ] ,a 1 ( l ) is monic with deg A [ ~ ν 1 ; ~ ν 2 ] ,a 1 ( l ) = ν 1 ,a 1 ( l ) ( l ) − 1 = k 1 ( l + 1) − 1, so that we are dealing with a type I I normalization and therefore we can write A ( l ) a = A (II ,a 1 ( l )) [ ~ ν 1 ; ~ ν 2 ] ,a . Dual equations to (36) are l X j =0 g i,j ¯ S 0 j,l = 0 , i = 0 , 1 , . . . , l − 1 , (38) l X j =0 g l,j ¯ S 0 j,l = 1 . (39) No w, using again (18) and (35), (38) b ecomes Z  p 2 X b =1 ¯ A ( l ) b ( x ) w 2 ,b ( x )  w 1 ,a ( x ) x k d µ ( x ) = 0 , deg ¯ A ( l ) b ≤ ν 2 ,b ( l ) − 1 , (40) 0 ≤ k ≤ ν 1 ,a ( l − 1) − 1 , 15 while (39) reads Z R  p 2 X b =1 ¯ A ( l ) b ( x ) w 2 ,b ( x )  w 1 ,a 1 ( l ) ( x ) x k 1 ( l ) d µ ( x ) = 1 , (41) where using (19) we obtain k 1 ( l ) = ν 1 ,a 1 ( l ) ( l − 1) . (42) As ab o ve we are dealing with m ultiple orthogonal p olynomials and therefore ¯ A ( l ) b = ¯ A [ ~ ν 2 ; ~ ν 1 ] ,b , with ~ ν 1 = ~ ν 1 ( l − 1) and ~ ν 2 = ~ ν 2 ( l ), which now happ ens to ha ve a normalization of type I and consequently w e write ¯ A ( l ) b = ¯ A (I ,a 1 ( l )) [ ~ ν 2 ; ~ ν 1 ] ,b . Giv en a deﬁnite sign ﬁnite Borel measure the corresp onding set of monic orthogonal p olynomials { p l } ∞ l =0 , deg p l = l , can b e viewed as a ladder of p olynomials, in which to get up to a given degree one needs to ascend l steps in the ladder. F or m ultiple orthogonalit y the situation is diﬀerent as w e ha v e, instead of a chain, a m ulti-dimensional lattice of degrees. Let us consider a perfect combination ( µ, ~ w 1 , ~ w 2 ) and the corresp onding set of m ultiple orthogonal p olynomials { A [ ~ ν 1 ; ~ ν 2 ] ,a } p 1 a =1 , with degree vectors such that | ~ ν 1 | = | ~ ν 2 | + 1. There alwa ys exists comp ositions ~ n 1 , ~ n 2 and an integer l with | ~ ν 1 | = l + 1 and | ~ ν 2 | = l such that the p olynomials { A ( l ) a } p 1 a =1 coincides with { A [ ~ ν 1 ; ~ ν 2 ] ,a } p 1 a =1 . Therefore, the set of sets of multiple orthogonal p olynomials  { A ( k ) a } p 1 a =1 , k = 0 , . . . , l  , can b e understo od as a ladder leading to the desired set of m ultiple orthogonal p olynomials { A [ ~ ν 1 ; ~ ν 2 ] ,a } p 1 a =1 after ascending l steps in the ladder, very muc h in same st yle as in standard orthogonalit y (non multiple) setting. The ladder can b e identiﬁed with the comp ositions ( ~ n 1 , ~ n 2 ). Ho wev er, by no means there is alwa ys a unique ladder to achiev e this, in general there are several comp ositions that do the job. A particular ladder, which we refer to as the simplest [ ~ ν 1 ; ~ ν 2 ] ladder, is given by the choice ~ n 1 = ~ ν 1 and ~ n 2 = ~ ν 2 + ~ e 2 ,p 2 . Many of the expressions that will be deriv ed later on in this pap er for m ultiple orthogonal p olynomials and second kind functions only dep end on the integers ( ~ ν 1 , ~ ν 2 ) and not on the particular ladder chosen, and therefore comp ositions, one uses to reach to it. 2.4 Linear forms and m ultiple bi-orthogonalit y W e in tro duce linear forms asso ciated with multiple orthogonal p olynomials as follo ws Deﬁnition 5. Strings of line ar forms and dual line ar forms asso ciate d with multiple orto gonal p olynomials and their duals ar e deﬁne d by Q :=    Q (0) Q (1) . . .    = S ξ 1 , ¯ Q :=    ¯ Q (0) ¯ Q (1) . . .    = ( ¯ S − 1 ) > ξ 2 , (43) It can b e immediately chec ked that Prop osition 4. The line ar forms and their duals, intr o duc e d in Deﬁnition 5, ar e given by Q ( l ) ( x ) := p 1 X a =1 A ( l ) a ( x ) w 1 ,a ( x ) , ¯ Q ( l ) ( x ) := p 2 X b =1 ¯ A ( l ) b ( x ) w 2 ,b ( x ) . (44) Sometimes w e use the alternativ e notation Q ( l ) = Q [ ~ ν 1 ; ~ ν 2 ] and ¯ Q ( l ) = ¯ Q [ ~ ν 2 ; ~ ν 1 ] . It is also trivial to chec k the follo wing Prop osition 5. The ortho gonality r elations Z Q ( l ) ( x ) w 2 ,b ( x ) x k d µ ( x ) = 0 , 0 ≤ k ≤ ν 2 ,b ( l − 1) − 1 , b = 1 , . . . , p 2 , Z ¯ Q ( l ) ( x ) w 1 ,a ( x ) x k d µ ( x ) = 0 , 0 ≤ k ≤ ν 1 ,a ( l − 1) − 1 , a = 1 , . . . , p 1 , (45) ar e fulﬁl le d. Moreo ver, we hav e that these linear forms are bi-orthogonal 16 Prop osition 6. The fol lowing multiple bi-ortho gonality r elations among line ar forms and their duals Z Q ( l ) ( x ) ¯ Q ( k ) ( x )d µ ( x ) = δ l,k , l , k ≥ 0 , (46) hold. Pr o of. Observe that Z R Q ( x ) ¯ Q ( x ) > d µ ( x ) = Z S ξ 1 ( x ) ξ 2 ( x ) > ¯ S − 1 d µ ( x ) from (43) = S  Z ξ 1 ( x ) ξ 2 ( x ) > d µ ( x )  ¯ S − 1 = S g ¯ S − 1 from (10) = I . from (33) Deﬁnition 6. Denote by ξ [ l ] i , i = 1 , 2 the trunc ate d ve ctor forme d with the ﬁrst l c omp onents of ξ i . W e are ready to give diﬀerent expressions for these linear forms and their duals Prop osition 7. The line ar forms c an b e expr esse d in terms of the moment matrix in the fol lowing diﬀer ent ways Q ( l ) = ξ ( l ) 1 −  g l, 0 g l, 1 · · · g l,l − 1  ( g [ l ] ) − 1 ξ [ l ] 1 = ¯ S l,l  0 0 · · · 0 1  ( g [ l +1] ) − 1 ξ [ l +1] 1 = 1 det g [ l ] det           g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 ξ (0) 1 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 ξ (1) 1 . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 ξ ( l − 1) 1 g l, 0 g l, 1 · · · g l,l − 1 ξ ( l ) 1           , l ≥ 1 , (47) and the dual line ar forms as ¯ Q ( l ) = ( ¯ S l,l ) − 1  ξ ( l ) 2 − ( ξ [ l ] 2 ) > ( g [ l ] ) − 1      g 0 ,l g 1 ,l . . . g l − 1 ,l       = ( ξ [ l +1] 2 ) > ( g [ l +1] ) − 1        0 0 . . . 0 1        = 1 det g [ l +1] det        g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 ,l . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 ,l ξ (0) 2 ξ (1) 2 · · · ξ ( l − 1) 2 ξ ( l ) 2        , l ≥ 0 . (48) Pr o of. See App endix A. As a consequence we get diﬀerent expressions for the multiple orthogonal p olynomials and their duals 17 Corollary 1. The multiple ortho gonal p olynomials and their duals have the fol lowing alternative expr essions A ( l ) a = χ ( l ) 1 ,a −  g l, 0 g l, 1 · · · g l,l − 1  ( g [ l ] ) − 1 χ [ l ] 1 ,a = ¯ S l,l  0 0 . . . 0 1  ( g [ l +1] ) − 1 χ [ l +1] 1 ,a = 1 det g [ l ] det            g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 χ (0) 1 ,a g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 χ (1) 1 ,a . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 χ ( l − 1) 1 ,a g l, 0 g l, 1 · · · g l,l − 1 χ ( l ) 1 ,a ,            , l ≥ 1 . (49) and ¯ A ( l ) b = ( ¯ S l,l ) − 1  χ ( l ) 2 ,b − ( χ [ l ] 2 ,b ) > ( g [ l ] ) − 1      g 0 ,l g 1 ,l . . . g l − 1 ,l       (50) = ( χ [ l +1] 2 ,b ) > ( g [ l +1] ) − 1        0 0 . . . 0 1        = 1 det g [ l +1] det         g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 ,l . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 ,l χ (0) 2 ,b χ (1) 2 ,b . . . χ ( l − 1) 2 ,b χ ( l ) 2 ,b         , l ≥ 0 . (51) Observ e that (208), App endix A, implies ¯ S l,l = det g [ l +1] det g [ l ] . (52) 2.5 F unctions of the second kind The Cauch y transforms of the linear forms (44) play a crucial role in the Riemann–Hilb ert problem associated with the multiple orthogonal p olynomials of mixed t yp e [14]. F ollowing the approac h of Adler and v an Mo erb ek e we will sho w that these Cauch y transforms are also related to the LU factorization considered in this pap er. Observ e that the construction of multiple orthogonal p olynomials p erformed so far is synthesized in the following strings of multiple orthogonal p olynomials and their duals A a :=     A (0) a A (1) a . . .     = S χ 1 ,a , ¯ A b :=     ¯ A (0) b ¯ A (1) b . . .     = ( ¯ S − 1 ) > χ 2 ,b , a = 1 , . . . , p 1 , b = 1 , . . . , p 2 . (53) In order to complete these formulae and in terms of χ ∗ as in (8) we consider Deﬁnition 7. L et us intr o duc e the fol lowing formal semi-inﬁnite ve ctors C b =     C (0) b C (1) b . . .     = ¯ S χ ∗ 2 ,b ( z ) , ¯ C a =     ¯ C (0) a ¯ C (1) a . . .     = ( S − 1 ) > χ ∗ 1 ,a ( z ) , b = 1 , . . . , p 2 , a = 1 , . . . , p 1 , (54) that we c al l strings of se c ond kind functions. 18 These ob jects are actually Cauc hy transforms of the linear forms Q ( l ) , l ∈ Z + , whenev er the series conv erge and outside the supp ort of the measures inv olved. Notice that ﬁxed z ∈ C the en tries in each string ¯ C a and C b are series not necessarily con v ergent. In the non-con v ergent case we ob viously understand the deﬁnition only formally . F or eac h l ∈ Z + w e denote by ¯ D ( l ) a and D ( l ) b on C the domains where the series ¯ C ( l ) a and C ( l ) b are uniform con vergen t, resp ectiv ely , and we understand them as their corresp onding limits. F rom prop erties of T aylor’s series, we know that uniform conv ergence of these series hops only on ¯ D ( l ) a and D ( l ) b when they are the biggest open disks around z = ∞ whic h do not contain the respectively supp orts, supp( w 2 ,a d µ ) and supp( w 2 ,b d µ ). Outside the sets ¯ D ( l ) a and D ( l ) b the series div erges at ev ery point. Hence to ha v e non-empty sets in ¯ D ( l ) a and D ( l ) b the corresponding supp orts supp( w 2 ,a d µ ) and supp( w 2 ,b d µ ) must b e b ounded. Prop osition 8. F or e ach l ∈ Z + the se c ond kind functions c an b e expr esse d as fol lows C ( l ) b ( z ) = Z Q ( l ) ( x ) w 2 ,b ( x ) z − x d µ ( x ) , z ∈ D ( l ) b \ supp( w 1 ,b d µ ( x )) , ¯ C ( l ) a ( z ) = Z ¯ Q ( l ) ( x ) w 1 ,a ( x ) z − x d µ ( x ) , z ∈ ¯ D ( l ) a \ supp( w 2 ,a d µ ( x )) . (55) Pr o of. The Gauss–Borel factorization leads to C ( l ) b ( z ) = ∞ X n =0 l X k =0 S lk g kn (Π 2 ,b χ ∗ 2 ( z )) n = ∞ X n =0 Z l X k =0 S lk x k 1 ( k ) w 1 ,a 1 ( k ) ( x ) w 2 ,b ( x ) x n z n +1 d µ ( x ) use (12) = ∞ X n =0 1 z n +1 Z x n Q ( l ) ( x ) w 2 ,b ( x )d µ ( x ) . use (44) When D ( l ) b \ supp( w 2 ,b d µ ) = ∅ the pro of is trivial. Giv en a non empty compact set K ⊂ D ( l ) b \ supp( w 2 ,b d µ ) 6 = ∅ and re- calling the closed character of supp( w 2 ,b d µ ), we hav e that the distance b etw e en them d ( l ) b ( K ) := distance( K , supp( w 2 ,b d µ )) > 0 is p ositiv e and that sup {| z | : z ∈ K} =: M K < + ∞ . T aking into account that the series C ( l ) b ( z ) = ∞ X n =0 1 z n +1 Z x n Q ( l ) ( x ) w 2 ,b ( x )d µ ( x ) con verges uniformly on K we can ensure lim n →∞ sup | z |∈K      1 z n +1 Z x n Q ( l ) ( x ) w 2 ,b ( x )d µ ( x )      = 0 . (56) Hence, we hav e the bound      n X i =0 1 z i +1 Z x i Q ( l ) ( x ) w 2 ,b ( x )d µ ( x ) − Z Q ( l ) ( x ) w 2 ,b ( x ) 1 z − x d µ ( x )      =     z 1 z n +1 Z x n Q ( l ) ( x ) w 2 ,b ( x ) d µ ( x ) z − x     ≤ ≤ M K d ( l ) b ( K ) sup | z |∈K      1 z n +1 Z x n Q ( l ) ( x ) w 2 ,b ( x )d µ ( x )      , ∀ z ∈ K . (57) T aking into accoun t (56) we deduce from (57) the ﬁrst equality for an y compact set K . Therefore, we get the ﬁrst claim of the Prop osition; the second equality can be prov ed analogously . Giv en l ≥ 1 and a = 1 , · · · , p the + ( − ) asso ciated integer is the smallest (largest) integer l + a ( l − a ) such that l + a ≥ l ( l − a ≤ l ) and a ( l + a ) = a ( a ( l − a ) = a ). It can be shown that l − a :=      q ( l ) | ~ n | + P a i =1 n i − 1 , a < a ( l ) , l, a = a ( l ) , q ( l ) | ~ n | − P p i = a +1 n i − 1 , a > a ( l − 1) , l + a :=      ( q ( l ) + 1) | ~ n | + P a − 1 i =1 n i , a < a ( l ) , l, a = a ( l ) , ( q ( l ) + 1) | ~ n | − P p i = a n i , a > a ( l ) . (58) 19 T o giv e a determinantal expression for these second kind formal series we need Deﬁnition 8. We intr o duc e Γ ( l ) k,a := ∞ X k 0 = l + a g k 0 ,k z − k 1 ( k 0 ) − 1 δ a 1 ( k 0 ) ,a , ¯ Γ ( l ) k,b := ∞ X k 0 = ¯ l + b g k,k 0 z − k 2 ( k 0 ) − 1 δ a 2 ( k 0 ) ,b . (59) Her e l + a is the + asso ciate d inte ger within the ~ n 1 c omp osition, while ¯ l + b is the + asso ciate d inte ger for the ~ n 2 c omp osition. With these deﬁnitions we can state Prop osition 9. The fol lowing determinantal expr essions for the functions of the se c ond kind hold C ( l ) b = 1 det g [ l ] det             g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 ¯ Γ ( l ) 0 ,b g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 ¯ Γ ( l ) 1 ,b . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 ¯ Γ ( l ) l − 1 ,b g l, 0 g l, 1 · · · g l,l − 1 ¯ Γ ( l ) l,b ,             , l ≥ 1 , (60) ¯ C ( l ) a = 1 det g [ l +1] det         g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 ,l . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 ,l Γ ( l ) 0 ,a Γ ( l ) 1 ,a · · · Γ ( l ) l − 1 ,a Γ ( l ) l,a         , l ≥ 1 . (61) Pr o of. See App endix A. F ollowing [20] we consider the Marko v–Stieltjes functions and p olynomials of the second type. Deﬁnition 9. The Markov–Stieltjes functions ar e deﬁne d by ˆ µ a,b ( z ) := Z w 1 ,a ( x ) w 2 ,b ( x ) z − x d µ ( x ) , (62) in terms of which we deﬁne H ( l ) b ( z ) := p 1 X a =1 A ( l ) a ( z ) ˆ µ a,b ( z ) − C ( l ) b ( z ) , ¯ H ( l ) a ( z ) := p 2 X b =1 ˆ µ a,b ( z ) ¯ A ( l ) b ( z ) − ¯ C ( l ) a ( z ) . (63) Prop osition 10. The functions H ( l ) b and ¯ H ( l ) a ar e p olynomials in z . Pr o of. The reader should notice that the functions H ( l ) b and ¯ H ( l ) a are H ( l ) b ( z ) = Z p 1 X a =1 w 1 ,a ( x ) A ( l ) a ( z ) − A ( l ) a ( x ) z − x w 2 ,b ( x )d µ ( x ) , ¯ H ( l ) a ( z ) = Z p 2 X b =1 w 1 ,a ( x ) ¯ A ( l ) b ( z ) − ¯ A ( l ) b ( x ) z − x w 2 ,b ( x )d µ ( x ) , and as z = x is a zero of the p olynomials A ( l ) a ( z ) − A ( l ) a ( x ) and ¯ A ( l ) b ( z ) − ¯ A ( l ) b ( x ) from the ab o v e formulae we conclude that they are indeed p olynomials in z . 20 2.6 Recursion relations The moment matrix has a Hankel t yp e symmetry that implies the recursion relations and the Christoﬀel–Darb oux form ula. W e consider the shift op erators deﬁned b y Λ a := ∞ X k =0 e a ( k ) e a ( k + 1) > . (64) notice that • Λ a lea ves inv ariant the subspaces Π a 0 R ∞ , for a 0 = 1 , . . . , p , and Π a 0 Λ a = Λ a Π a 0 . • The set of semi-inﬁnite matrices { Λ j a } a =1 ,...,p j =1 , 2 ,... is commutativ e. • W e hav e the eigen v alue prop ert y Λ a χ a 0 = δ a,a 0 z χ a . (65) Deﬁnition 10. We deﬁne the fol lowing multiple shift matric es Υ 1 := p 1 X a =1 Λ 1 ,a , Υ 2 := p 2 X b =1 Λ 2 ,b , (66) and we also intr o duc e the inte gers N 1 ,a := | ~ n 1 | − n 1 ,a + 1 = X a 0 =1 ,...,p 1 a 0 6 = a n 1 ,a 0 + 1 , a = 1 , . . . , p 1 , N 1 := max a =1 ,...,p 1 N 1 ,a , N 2 ,b := | ~ n 2 | − n 2 ,b + 1 = X b 0 =1 ,...,p 2 b 0 6 = b n 2 ,b 0 + 1 , b = 1 , . . . , p 2 , N 2 := max b =1 ,...,p 2 N 2 ,b . A careful but straightforw ard c omputation leads to Prop osition 11. We have the fol lowing structur e for Υ 1 and Υ 2 Υ 1 = D 1 , 0 Λ + D 1 , 1 Λ N 1 , 1 + · · · + D 1 ,p 1 Λ N 1 ,p 1 , Υ 2 = D 2 , 0 Λ + D 2 , 1 Λ N 2 , 1 + · · · + D 2 ,p 2 Λ N 2 ,p 2 . wher e D 1 ,a , a = 1 , . . . , p 1 , and D 2 ,b , b = 1 , . . . , p 2 , ar e the fol lowing semi-inﬁnite diagonal matric es: D 1 ,a = diag( D 1 ,a (0) , D 1 ,a (1) , . . . ) , D 1 ,a ( n ) := ( 1 , n = k | ~ n 1 | + P a a 0 =1 n 1 ,a 0 − 1 , k ∈ Z + , 0 , n 6 = k | ~ n 1 | + P a a 0 =1 n 1 ,a 0 − 1 , k ∈ Z + , D 1 , 0 = I − p 1 X a =1 D 1 ,a , D 2 ,b = diag( D 2 ,b (0) , D 2 ,b (1) , . . . ) , D 2 ,b ( n ) := ( 1 , n = k | ~ n 2 | + P b b 0 =1 n 2 ,b 0 − 1 , k ∈ Z + , 0 , n 6 = k | ~ n 2 | + P b b 0 =1 n 2 ,b 0 − 1 , k ∈ Z + , D 2 , 0 = I − p 1 X b =1 D 2 ,b . In terms of these shift matrices we can describ e the particular Hankel symmetries for the m omen t matrix Prop osition 12. The moment matrix g satisﬁes the Hankel typ e symmetry Υ 1 g = g Υ > 2 . (67) Pr o of. With the use of (19) and (12) we get Λ 1 ,a g Π 2 ,b = Π 1 ,a g Λ > 2 ,b , (68) and summing up in a = 1 , . . . , p 1 and b = 1 , . . . , p 2 w e get the desired result. Observ e that from (67) we deduce that in spite of being all the truncated moment matrices g [ l ] , l = 1 , 2 , . . . in vertible, the moment matrix g = lim l →∞ g [ l ] is not inv ertible. Supp ose that the inv erse g − 1 = ( ˜ g i,j ) 1 ,j =0 , 1 ,... of g exists so that (67) implies g − 1 Υ 1 = Υ > 2 g − 1 , and therefore ˜ g i, 0 = ˜ g 0 ,j = 0 for all i, j = 0 , 1 , . . . , which is con tradictory with the inv ertibility of g . 21 Prop osition 13. F r om the symmetry of the moment matrix one derives S Υ 1 S − 1 = ¯ S Υ > 2 ¯ S − 1 . (69) Pr o of. If we introduce (33) in to (67) we get Υ 1 S − 1 ¯ S = S − 1 ¯ S Υ > 2 ⇒ S Υ 1 S − 1 = ¯ S Υ > 2 ¯ S − 1 . Deﬁnition 11. We deﬁne the matric es J := J + + J − , J + := ( S Υ 1 S − 1 ) + , J − := ( ¯ S Υ > 2 ¯ S − 1 ) − , wher e the sub-indic es + and − denotes the upp er triangular and strictly lower triangular pr oje ctions. Th us, J + is an upp er triangular matrix and J − a strictly low er triangular matrix. Moreo v er, from the string equation (69) we hav e the alternative expressions J = S Υ 1 S − 1 = ¯ S Υ > 2 ¯ S − 1 . W e no w analyze the structure of J + := ( S D 1 , 0 Λ S − 1 ) + + ( S D 1 , 1 Λ N 1 , 1 S − 1 ) + + · · · + ( S D 1 ,p 1 Λ N 1 ,p 1 S − 1 ) + . It is clear that we need to ev aluate expressions of the form S E i,j S − 1 with i = κ 1 ( k , a ) − 1 and j = κ 1 ( k + 1 , a − 1) b eing κ 1 ( k , a ) := k | ~ n 1 | + P a a 0 =1 n 1 ,a 0 . Given the form of S , see (33), we hav e ( S E i,j S − 1 ) + = E i,j + X l,l ∈ L i,j s l,l 0 E l,l 0 , L i,j := { ( l, l 0 ) ∈ Z 2 + | l < i, l 0 < j, l 0 ≥ l } , for some num b ers s l,l 0 ∈ R dep ending on the co eﬃcien ts of S and on i, j ; this matrix has zero es everywhere but on a region of it that can be represented as a righ t triangle with hypotenuse lying on the main diagonal, this h yp oten use has its opp osite v ertex precisely on the ( i, j ) p osition. Therefore J + = ( S D 1 , 0 Λ S − 1 ) + + p 1 X a =1 ∞ X k =0 E κ 1 ( k,a ) − 1 ,κ 1 ( k +1 ,a − 1) + X l,l 0 ∈ L 1 ,k,a s l,l 0 E l,l 0 ! , L 1 ,k,a := L κ 1 ( k,a ) − 1 ,κ 1 ( k +1 ,a − 1) W e see that J + can b e schematically represented as a staircase, the ~ n 1 -staircase, descending o ver the main diagonal with steps –which are built with right triangles with h yp oten use lying on the main diagonal and opp osite vertex (and therefore corner of the step) lo cated at the ( κ 1 ( k , a ) − 1 , κ 1 ( k + 1 , a − 1)) p osition of the matrix– having width and height given by the integers in the comp osition ~ n 1 . F or example, the j -th step has width n 1 , j p 1 − [ j p 1 ] and height n 1 , j +1 p 1 − [ j +1 p 1 ] . A similar description holds for J > − but replacing the comp osition ~ n 1 b y ~ n 2 . Therefore, the matrix J is a generalized Jacobi matrix and, in con trast with the non m ultiple case, now is m ulti-diagonal (ha ving in general more than three diagonals) and has a diagonal band of length N 1 + N 2 + 1. Moreov er, this band has a num b er of zero es on it, according to the ~ n 1 -stair on the upp er part and to the ~ n 2 -stair in the low er part, we refer to this as a double ( ~ n 1 , ~ n 2 )-staircase shap e. T o illustrate this snake shape let us write for the case ~ n 1 = (4 , 3 , 2) and ~ n 2 = (3 , 2) 22 the corresp onding truncated, l = 27, Jacobi type matrix J [27] =                                              ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗                                              , (70) where ∗ denotes a non-necessarily null real n umber. W e can write J = J N 1 Λ N 1 + · · · + J 1 Λ + J 0 + J − 1 Λ > + · · · + J − N 2 (Λ > ) N 2 , (71) where J i = diag( J i (0) , J i (1) , . . . ). F or conv enience we extend the notation with J r ( s ) = 0 whenever r + s < 0 or s < 0. W e in tro duce Deﬁnition 12. The semi-inﬁnite ve ctors c b and ¯ c a ar e given by c b := ¯ S e 2 ,b (0) , b = 1 , . . . , p 2 , ¯ c a := ( S − 1 ) > e 1 ,a (0) , a = 1 , . . . , p 1 . (72) It is not diﬃcult to show that c b = n 2 , 1 + ··· + n 2 ,b − 1 X l =0 ¯ S l,n 2 , 1 + ··· + n 2 ,b − 1 e l , ¯ c a = n 1 , 1 + ··· + n 1 ,a − 1 X l =0 ( S − 1 ) n 1 , 1 + ··· + n 1 ,a − 1 ,l e l . (73) The semi-inﬁnite matrices J and J > ha ve the following imp ortan t property Prop osition 14. The fol lowing e quations ar e fulﬁl le d J A a ( z ) = z A a ( z ) , J > ¯ A b ( z ) = z ¯ A b ( z ) , J C b ( z ) = z C b ( z ) − c b , J > ¯ C a ( z ) = z C a ( z ) − ¯ c a . (74) Pr o of. F rom (53) and (54) J A a ( z ) = S Υ 1 S − 1 S χ 1 ,a ( z ) = z S χ 1 ,a = z A a ( z ) , J C b ( z ) = ¯ S Υ > 2 ¯ S − 1 ¯ S χ ∗ 2 ,b ( z ) = ¯ S ( z χ ∗ 2 ,b ( z ) − e b (0)) = z C b ( z ) − c b , where we hav e taken in to accoun t that Υ > 2 χ ∗ 2 ,b ( z ) = z χ ∗ 2 ,b ( z ) − e b (0). F or J > w e pro ceed similarly: J > ¯ A b ( z ) = ( ¯ S − 1 ) > Υ 2 ¯ S > ( ¯ S − 1 ) > χ 2 ,b ( z ) = z ( ¯ S − 1 ) > χ 2 ,a = z ¯ A b ( z ) , J > ¯ C a ( z ) = ( S − 1 ) > Υ > 1 S > ( S − 1 ) > χ ∗ 1 ,a ( z ) = ( S − 1 ) > ( z χ ∗ 1 ,a ( z ) − e a (0)) = z ¯ C a ( z ) − ¯ c a . 23 Theorem 2. The multiple ortho gonal p olynomials and their asso ciate d se c ond kind functions fulﬁl l the fol lowing r e cursion r elations z A ( l ) a ( z ) = J − N 2 ( l ) A ( l − N 2 ) a ( z ) + · · · + J N 1 ( l ) A ( l + N 1 ) a ( z ) , z C ( l ) b ( z ) − c ( l ) b = J − N 2 ( l ) C ( l − N 2 ) b ( z ) + · · · + J N 1 ( l ) C ( l + N 1 ) b ( z ) , (75) while the dual r elations ar e z ¯ A ( l ) b ( z ) = J − N 2 ( l + N 2 ) ¯ A ( l + N 2 ) b ( z ) + · · · + J N 1 ( l − N 1 ) ¯ A ( l − N 1 ) b ( z ) , z ¯ C ( l ) a ( z ) − ¯ c a = J − N 2 ( l + N 2 ) ¯ C ( l + N 2 ) a ( z ) + · · · + J N 1 ( l − N 1 ) ¯ C ( l − N 1 ) a ( z ) . (76) W e see that given in tegers ( ~ ν 1 , ~ ν 2 ) there are several recursion relations associated with A [ ~ ν 1 ; ~ ν 2 ] ,a . In fact they are as man y as diﬀerent ladders exists leading to this set of degrees. F or the simplest ladder, i.e. ~ n 1 = ~ ν 1 and ~ n 2 = ~ ν 2 + ~ e 2 ,p 2 , w e get the longest recursion, in the sense that w e hav e more p olynomials contributing in the recursion relation, as smaller are the integers in the comp ositions shorter is the recursion. Observ e also that the m ultiple orthogonal p olynomials in volv ed in each case are diﬀeren t. A ttending to (70) we get that the recursion relations corresp onding to l = 8 an l = 14 are of the form z A (8) a ( z ) = ∗ A (4) a ( z ) + · · · + ∗ A (15) a ( z ) + A (16) a ( z ) , a = 1 , 2 , 3 , z A (14) a ( z ) = ∗ A (12) a ( z ) + · · · + ∗ A (18) a ( z ) , a = 1 , 2 , 3 . W e see that the ﬁrst recursion has 13 terms while the second one only 7 terms. In order to iden tify these p olynomials with mops of the form A [ ~ ν 1 ; ~ ν 2 ,a ] w e use follo wing the table of degrees for the comp ositions ~ n 1 = (4 , 3 , 2) and ~ n 2 = (3 , 2) is l 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ~ ν 1 ( l ) (4,0,0) (4,1,0) (4,2,0) (4,3,0) (4,3,1) (4,3,2) (5,3,2) (6,3,2) (7,3,2) (8,3,2) (8,4,2) (8,5,2) (8,6,2) (8,6,3) (8,6,4) ~ ν 2 ( l − 1) (2,0) (3,0) (3,1) (3,2) (4,2) (5,2) (6,2) (6,3) (6,4) (7,4) (8,4) (9,4) (9,5) (9,6) (10,6) 2.7 Christoﬀel–Darb oux type form ulae F rom (53) and (54) we can infer the v alue of the following series constructed in terms of multiple orthogonal p olynomials and corresp onding functions of the second kind. Prop osition 15. The fol lowing r elations hold ∞ X l =0 ¯ C ( l ) a ( z ) A ( l ) a 0 ( z 0 ) = δ a,a 0 z − z 0 , | z 0 | < | z | , ∞ X l =0 C ( l ) b ( z ) ¯ A ( l ) b 0 ( z 0 ) = δ b,b 0 z − z 0 , | z 0 | < | z | , ∞ X l =0 ¯ C ( l ) a ( z ) C ( l ) b ( z 0 ) = − ˆ µ a,b ( z ) − ˆ µ a,b ( z 0 ) z − z 0 , | z | , | z 0 | > R a,b , (77) wher e R a,b is the r adius of any origin c enter e d disk c ontaining supp( w 1 ,a w a,b d µ ) . Pr o of. See App endix A. 2.7.1 Pro jection op erators and the Christoﬀel–Darb oux k ernel T o in tro duce the Christoﬀel–Darb oux kernel we need Deﬁnition 13. We wil l use the fol lowing sp ans H [ l ] 1 = R { ξ (0) 1 , . . . , ξ ( l − 1) 1 } , H [ l ] 2 = R { ξ (0) 2 , . . . , ξ ( l − 1) 2 } , (78) and their limits H 1 = n X 0 ≤ l ∞ c l ξ ( l ) 1 , c l ∈ R o , H 2 = n X 0 ≤ l ∞ c l ξ ( l ) 2 , c l ∈ R o . (79) 24 The c orr esp onding splittings H 1 = H [ l ] 1 ⊕ ( H [ l ] 1 ) ⊥ , H 2 = H [ l ] 2 ⊕ ( H [ l ] 2 ) ⊥ , (80) induc e the asso ciate d ortho gonal pr oje ctions π ( l ) 1 : H 1 → H [ l ] 1 , π ( l ) 2 : H 2 → H [ l ] 2 . (81) In the previous deﬁnition l  ∞ means that in the series there are only a ﬁnite n umber of nonzero con tributions. It is easy to realize that Prop osition 16. We have the fol lowing char acterization of the pr evious line ar subsp ac es H [ l ] 1 = R { Q (0) , . . . , Q ( l − 1) } , H [ l ] 2 = R { ¯ Q (0) , . . . , ¯ Q ( l − 1) } , ( H [ l ] 1 ) ⊥ = n X l ≤ j ∞ c j Q ( j ) , c j ∈ R o , ( H [ l ] 2 ) ⊥ = n X l ≤ j ∞ c j ¯ Q ( j ) , c j ∈ R o , (82) and H 1 = n X 0 ≤ l ∞ c l Q ( l ) , c l ∈ R o , H 2 = n X 0 ≤ l ∞ c l ¯ Q ( l ) , c l ∈ R o . (83) Deﬁnition 14. The Christoﬀel–Darb oux kernel is K [ l ] ( x, y ) := l − 1 X k =0 Q ( k ) ( y ) ¯ Q ( k ) ( x ) . (84) This is the kernel of the integral represen tation of the pro jections introduced in Deﬁnition 13. Prop osition 17. The inte gr al r epr esentation ( π ( l ) 1 f )( y ) = Z K [ l ] ( x, y ) f ( x )d µ ( x ) , ∀ f ∈ H 1 , ( π ( l ) 2 f )( y ) = Z K [ l ] ( y , x ) f ( x )d µ ( x ) , ∀ f ∈ H 2 , (85) holds. Pr o of. It follows from the bi-orthogonalit y condition (46). This Christoﬀel–Darb oux kernel has the repro ducing prop ert y Prop osition 18. The kernel K [ l ] ( x, y ) fulﬁl ls K [ l ] ( x, y ) = Z K [ l ] ( x, v ) K [ l ] ( v , y )d µ ( v ) . (86) Pr o of. F rom f ( y ) = Z K [ l ] ( x, y ) f ( x )d µ ( x ) , ∀ f ∈ H [ l ] 1 , f ( y ) = Z K [ l ] ( y , x ) f ( x )d µ ( x ) , ∀ f ∈ H [ l ] 2 , and K [ l ] ( x, y ) ∈ H [ l ] 1 as a function of y and K [ l ] ( x, y ) ∈ H [ l ] 2 as a function of x w e conclude the repro ducing prop ert y . 25 2.7.2 The ABC t yp e theorem W e also hav e an ABC (Aitk en–Berg–Collar) t yp e theorem —here w e follo w [35]— for the Christoﬀel–Darb oux kernel Deﬁnition 15. The p artial Christoﬀel–Darb oux kernels ar e deﬁne d by K [ l ] b,a ( x, y ) := l − 1 X k =0 ¯ A ( k ) b ( x ) A ( k ) a ( y ) . (87) Observ e that K [ l ] ( x, y ) = X a =1 ,...,p 1 b =1 ,...,p 2 K [ l ] b,a ( x, y ) w 1 ,a ( y ) w 2 ,b ( x ) . (88) W e in tro duce the notation Deﬁnition 16. Any semi-inﬁnite ve ctor v c an b e written in blo ck form as fol lows v = v [ l ] v [ ≥ l ] ! , (89) wher e v [ l ] is the ﬁnite ve ctor forme d with the ﬁrst l c o eﬃcients of v and v [ ≥ l ] the semi-inﬁnite ve ctor forme d with the r emaining c o eﬃcients. This de c omp osition induc es the fol lowing blo ck structur e for any semi-inﬁnite matrix. g = g [ l ] g [ l, ≥ l ] g [ ≥ l,l ] g [ ≥ l ] ! . (90) F rom (33) we get Prop osition 19. Given a moment matrix g satisfying (33) we have g [ l ] = ( S [ l ] ) − 1 ¯ S [ l ] , (91) and ( S − 1 ) [ l ] = ( S [ l ] ) − 1 , ( ¯ S − 1 ) [ ≥ l ] = ( ¯ S [ ≥ l ] ) − 1 . Pr o of. Use the blo ck structure of g , S and ¯ S . Then, we are able to conclude the follo wing result Theorem 3. The Christoﬀel–Darb oux kernel is r elate d to the moment matrix in the fol lowing way K [ l ] b,a ( x, y ) = ( χ [ l ] 2 ,b ( x )) > ( g [ l ] ) − 1 χ [ l ] 1 ,a ( y ) . (92) Pr o of. The ABC theorem is a consequence of the follo wing chain of identities K [ l ] b,a ( x, y ) = (Π [ l ] ¯ A b ( x )) > (Π [ l ] A a ( y )) the sum is ov er the ﬁrst l comp onen ts = χ > 2 ,b ( x ) ¯ S − 1 Π [ l ] S χ 1 ,a ( y ) see (43) = χ > 2 ,b ( x )(Π [ l ] ¯ S − 1 Π [ l ] )(Π [ l ] S Π [ l ] ) χ 1 ,a ( y ) lo wer and upp er form of S and ¯ S = ( χ [ l ] 2 ,b ( x )) > ( ¯ S [ l ] ) − 1 S [ l ] χ [ l ] 1 ,a ( y ) = ( χ [ l ] 2 ,b ( x )) > ( g [ l ] ) − 1 χ [ l ] 1 ,a ( y ) LU factorization (33). W e immediately deduce the Corollary 2. F or the Christoﬀel–Darb oux kernel we have K [ l ] ( x, y ) = ( ξ [ l ] 2 ( x )) > ( g [ l ] ) − 1 ξ [ l ] 1 ( y ) . (93) 26 2.7.3 Christoﬀel–Darb oux form ula In this subsection we derive a Christoﬀel–Darb oux type formula from the symmetry prop ert y (67) of the momen t matrix g . W e need some preliminary lemmas Lemma 3. The r elations ( g [ l ] ) − 1 Υ [ l ] 1 − (Υ [ l ] 2 ) > ( g [ l ] ) − 1 = ( g [ l ] ) − 1  g [ l, ≥ l ] (Υ [ l, ≥ l ] 2 ) > − Υ [ l, ≥ l ] 1 g [ ≥ l,l ]  ( g [ l ] ) − 1 , (94) hold true. Pr o of. The ﬁrst blo ck of (67) is Υ [ l ] 1 g [ l ] + Υ [ l, ≥ l ] 1 g [ ≥ l,l ] = g [ l ] (Υ [ l ] 2 ) > + g [ l, ≥ l ] (Υ [ l, ≥ l ] 2 ) > , from where the result follows immediately . Lemma 4. We have Υ [ l ] ` ξ [ l ] ` ( x ) = xξ [ l ] ` ( x ) − Υ [ l, ≥ l ] ` ξ [ ≥ l ] ` ( x ) , ` = 1 , 2 . (95) Pr o of. It follows from the block decomp osition of Deﬁnitions 16 and the eigen-v alue prop ert y of Υ ` . After a careful computation from Deﬁnition 10 w e get Lemma 5. If we assume that l ≥ max( | ~ n 1 | , | ~ n 2 | ) we c an write Υ [ l, ≥ l ] 1 = p 1 X a =1 e ( l − 1) − a e > l + a − l , Υ [ l, ≥ l ] 2 = p 2 X b =1 e ( l − 1) − b e > ¯ l + b − l . (96) Her e l ± a is the ± asso ciate d inte ger within the ~ n 1 c omp osition, while ¯ l ± b is the ± asso ciate d inte ger for the ~ n 2 c omp osition. Finally , to derive a Christoﬀel–Darb oux type formula w e need the following ob jects Deﬁnition 17. Asso ciate d p olynomials ar e given by A ( l ) + a,a 0 ( y ) := χ ( l + a ) 1 ,a 0 ( y ) −  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y ) , ¯ A ( l ) − a,b 0 ( x ) := ( χ [ l +1] 2 ,b 0 ( x )) > ( g [ l +1] ) − 1 e l − a , A ( l ) − b,a 0 ( y ) := e > ¯ l − b ( g [ l +1] ) − 1 χ [ l +1] 1 ,a 0 ( y ) , ¯ A ( l ) + b,b 0 ( x ) :=  χ ( ¯ l + b ) 2 ,b 0 ( x ) − ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b       , (97) with the c orr esp onding line ar forms given by Q ( l ) + a := p 1 X a 0 =1 A ( l ) + a,a 0 w 1 ,a 0 , ¯ Q ( l ) − a := p 2 X b 0 =1 ¯ A ( l ) − a,b 0 w 2 ,b 0 , a = 1 , . . . , p 1 , Q ( l ) − b := p 1 X a 0 =1 A ( l ) − b,a 0 w 1 ,a 0 , ¯ Q ( l ) + b := p 2 X b 0 =1 ¯ A ( l ) + b,b 0 w 2 ,b 0 , b = 1 , . . . , p 2 . (98) Then, we can show that Theorem 4. Whenever l ≥ max( | ~ n 1 | , | ~ n 2 | ) the fol lowing Christoﬀel–Darb oux typ e formulae ( x − y ) K [ l ] a 0 ,b 0 ( x, y ) = p 2 X b =1 ¯ A ( l ) + b,b 0 ( x ) A ( l − 1) − b,a 0 ( y ) − p 1 X a =1 ¯ A ( l − 1) − a,b 0 ( x ) A ( l ) + a,a 0 ( y ) , ( x − y ) K [ l ] ( x, y ) = p 2 X b =1 ¯ Q ( l ) + b ( x ) Q ( l − 1) − b ( y ) − p 1 X a =1 ¯ Q ( l − 1) − a ( x ) Q ( l ) + a ( y ) , hold. 27 Pr o of. F rom Lemma 3 w e deduce ( χ [ l ] 2 ,b 0 ( x )) >  ( g [ l ] ) − 1 Υ [ l ] 1 − (Υ [ l ] 2 ) > ( g [ l ] ) − 1  χ [ l ] 1 ,a 0 ( y ) = ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1  g [ l, ≥ l ] (Υ [ l, ≥ l ] 2 ) > − Υ [ l, ≥ l ] 1 g [ ≥ l,l ]  ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y ) , so that, recalling Theorem 3, we get ( y − x ) K [ l ] b 0 ,a 0 ( x, y ) =( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1  g [ l, ≥ l ] (Υ [ l, ≥ l ] 2 ) > − Υ [ l, ≥ l ] 1 g [ ≥ l,l ]  ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y ) + ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1 Υ [ l, ≥ l ] 1 χ [ ≥ l ] 1 ,a 0 ( y ) − (Υ [ l, ≥ l ] 2 χ [ ≥ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1 ( χ [ l ] 1 ,a 0 ( y )) , (99) or ( x − y ) K ( l − 1) b 0 ,a 0 ( x, y ) =  ( χ [ ≥ l ] 2 ,b 0 ( x )) > − ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1 g [ l, ≥ l ]  (Υ [ l, ≥ l ] 2 ) > ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y ) − ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1 Υ [ l, ≥ l ] 1  χ [ ≥ l ] 1 ,a 0 ( y ) − g [ ≥ l,l ] ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y )  . Finally , from Lemma 5 we conclude Υ [ l, ≥ l ] 1 χ [ ≥ l ] 1 ,a 0 ( y ) = p 1 X a =1 e ( l − 1) − a χ ( l + a ) 1 ,a 0 ( y ) , Υ [ l, ≥ l ] 1 g [ ≥ l,l ] = p 1 X a =1 e ( l − 1) − a  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  , ( χ [ ≥ l ] 2 ,b 0 ( x )) > (Υ [ l, ≥ l ] 2 ) > = p 2 X b =1 χ ( ¯ l + b ) 2 ,b 0 ( x ) e > ( l − 1) − b , g [ l, ≥ l ] (Υ [ l, ≥ l ] 2 ) > = p 2 X b =1      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b      e > ( l − 1) − b , and consequently ( x − y ) K [ l ] b 0 ,a 0 ( x, y ) = p 2 X b =1  χ ( ¯ l + b ) 2 ,b 0 ( x ) − ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b       e > ( l − 1) − b ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y ) − p 1 X a =1 ( χ [ l ] 2 ,b 0 ( x )) > ( g [ l ] ) − 1 e ( l − 1) − a  χ ( l + a ) 1 ,a 0 ( y ) −  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  ( g [ l ] ) − 1 χ [ l ] 1 ,a 0 ( y )  . (100) Recalling Deﬁnition 17 we get the announced result. The asso ciated linear forms are identiﬁed with linear forms of m ultiple orthogonal p olynomials as follows Prop osition 20. We have the formulae Q ( l ) + a = Q (II ,a ) [ ~ ν 1 ( l − 1)+ ~ e 1 ,a ; ~ ν 2 ( l − 1)] , Q ( l ) − b = Q (I ,b ) [ ~ ν 1 ( l ); ~ ν 2 ( l ) − ~ e 2 ,b ] , ¯ Q ( l ) + b = ¯ Q (II ,b ) [ ~ ν 2 ( l − 1)+ ~ e 2 ,b ; ~ ν 1 ( l − 1)] , ¯ Q ( l ) − a = Q (I ,a ) [ ~ ν 2 ( l ); ~ ν 1 ( l ) − ~ e 1 ,a ] . (101) Pr o of. See App endix A. Prop osition 20 allows us to give the following form of the Christoﬀel–Darb oux formula stated in Theorem 4 28 Prop osition 21. F or l ≥ max( | ~ n 1 | , | ~ n 2 | ) the fol lowing ( x − y ) K [ l ] ( x, y ) = p 2 X b =1 ¯ Q (II ,b ) [ ~ ν 2 ( l − 1)+ ~ e 2 ,b ; ~ ν 1 ( l − 1)] ( x ) Q (I ,b ) [ ~ ν 1 ( l − 1); ~ ν 2 ( l − 1) − ~ e 2 ,b ] ( y ) − p 1 X a =1 ¯ Q (I ,a ) [ ~ ν 2 ( l − 1); ~ ν 1 ( l − 1) − ~ e 1 ,a ] ( x ) Q (II ,a ) [ ~ ν 1 ( l − 1)+ ~ e 1 ,a ; ~ ν 2 ( l − 1)] ( y ) . (102) ( x − y ) K [ l ] b 0 ,a 0 ( x, y ) = p 2 X b =1 ¯ A (II ,b ) [ ~ ν 2 ( l − 1)+ ~ e 2 ,b ; ~ ν 1 ( l − 1)] ,b 0 ( x ) A (I ,b ) [ ~ ν 1 ( l − 1); ~ ν 2 ( l − 1) − ~ e 2 ,b ] ,a 0 ( y ) − p 1 X a =1 ¯ A (I ,a ) [ ~ ν 2 ( l − 1); ~ ν 1 ( l − 1) − ~ e 1 ,a ] ,b 0 ( x ) A (II ,a ) [ ~ ν 1 ( l − 1)+ ~ e 1 ,a ; ~ ν 2 ( l − 1)] ,a 0 ( y ) . (103) holds. Relation (102) is precisely the Christoﬀel–Darb oux form ula deriv ed in [14], the diﬀerence here is that deriv ation is based on the Gauss–Borel factorization problem for the momen t matrix; i.e. only on algebraic arguments, and not in the Riemann–Hilb ert problem found in [14], and hence the conditions on the weigh ts are not so restrictive. Ho wev er, the reader should notice that the Christoﬀel–Darb oux kernel do es not dep end on the ladder determined by the composition vectors ~ n 1 , ~ n 2 , but only on the degree v ectors ~ ν 1 ( l − 1) and ~ ν 2 ( l − 1). This was noticed in [15] for t yp e I multiple orthogonality . Prop osition 22. The asso ciate d p olynomials intr o duc e d in Deﬁnition 17 have the fol lowing determinantal expr essions A ( l ) + a,a 0 = 1 det g [ l ] det            g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 χ (0) 1 ,a 0 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 χ (1) 1 ,a 0 . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 χ ( l − 1) 1 ,a 0 g l + a , 0 g l + a , 1 · · · g l + a ,l − 1 χ ( l + a ) 1 ,a 0            , (104) ¯ A ( l ) − a,b 0 ( x ) = ( − 1) l + l − a det g [ l +1] det              g 0 , 0 · · · g 0 ,l − 1 g 0 ,l . . . . . . . . . g l − a − 1 , 0 · · · g l − a − 1 ,l − 1 g l − a − 1 ,l g l − a +1 , 0 · · · g l − a +1 ,l − 1 g l − a +1 ,l . . . . . . . . . g l − 1 , 0 · · · g l − 1 ,l − 1 g l − 1 ,l χ (0) 2 ,b 0 · · · χ ( l − 1) 2 ,b 0 χ ( l ) 2 ,b 0              , (105) A ( l ) − b,a 0 = ( − 1) l + ¯ l − b det g [ l +1] det         g 0 , 0 · · · g 0 , ¯ l − b − 1 g 0 , ¯ l − b +1 · · · g 0 ,l − 1 χ (0) 1 ,a 0 . . . . . . . . . . . . . . . g l − 1 , 0 · · · g l − 1 , ¯ l − b − 1 g l − 1 , ¯ l − b +1 · · · g l − 1 ,l − 1 χ ( l − 1) 1 ,a 0 g l, 0 · · · g l, ¯ l − b − 1 g l, ¯ l − b +1 · · · g l,l − 1 χ ( l ) 1 ,a 0         , (106) ¯ A ( l ) + b,b 0 = 1 det g [ l ] det          g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 , ¯ l + b g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 , ¯ l + b . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 , ¯ l + b χ (0) 2 ,b 0 χ (1) 2 ,b 0 · · · χ ( l − 1) 2 ,b 0 χ ( ¯ l + b ) 2 ,b 0          . (107) 29 3 Connection with the multi-component 2D T o da Lattice hierarc h y In this section we introduce deformations of the Gauss–Borel factorization problem that giv e the connection with the theory of the integrable hierarchies of 2D T o da lattice t yp e, in the m ulti-comp onen t ﬂav or case. First, w e introduce the con tinuous ﬂows and then the discre te ones. Let us stress that b oth ﬂows could b e considered simultaneously but w e consider them separately for the sak e of simplicity and clearness in the exp osition. 3.1 Con tin uous deformations of the momen t matrix Deﬁnition 18. The deforme d moment matrix is given by g ~ n 1 ,~ n 2 ( t ) := W 0 ,~ n 1 ( t ) g ¯ W 0 ,~ n 2 ( t ) − 1 , (108) wher e we use the fol lowing semi-inﬁnite matric es W 0 ,~ n 1 ( t ) := p 1 X a =1 exp  ∞ X j =1 t j,a Λ j 1 ,a  ∈ G + , ¯ W 0 ,~ n 2 ( t ) := p 2 X b =1 exp  ∞ X j =1 ¯ t j,b (Λ > 2 ,b ) j  ∈ G − dep ending on t = ( t j,a , ¯ t j,b ) j,a,b with t j,a , ¯ t j,b ∈ R , j = 1 , 2 , . . . , a = 1 , . . . , p 1 and b = 1 , . . . , p 2 . As in the previous section and when the context is clear enough we will drop the subscripts asso ciated with the comp ositions ~ n 1 and ~ n 2 . The reader should notice that the follo wing semi-inﬁnite matrices are w ell deﬁned ( W 0 ,~ n 1 ( t ) − 1 ) > = p 1 X a =1 exp  − ∞ X j =1 t j,a (Λ > 1 ,a ) j  ∈ G − , ( ¯ W 0 ,~ n 2 ( t ) − 1 ) > = p 2 X b =1 exp  − ∞ X j =1 ¯ t j,b Λ j 2 ,b  ∈ G + . This deformation preserves the structure that characterizes g as a moment matrix, in fact w e hav e Theorem 5. The matrix g ( t ) is a moment matrix with new “deforme d weights” given by w 1 ,a ( x, t ) = E a ( x, t ) w 1 ,a ( x ) , E a := exp  ∞ X j =1 t j,a x j  , w 2 ,b ( x, t ) = ¯ E b ( x, t ) − 1 w 2 ,b ( x ) , ¯ E b := exp  ∞ X j =1 ¯ t j,b x j  . (109) Pr o of. Observe that W 0 ( t ) = X j ≥ 0 p 1 X a =1 σ ( a ) j ( t )Λ j 1 ,a , ¯ W 0 ( t ) − 1 = X j ≥ 0 p 2 X b =1 (Λ > 2 ,b ) j ¯ σ ( b ) j ( t ) , where σ ( a ) j is the j -th elemen tary Sc hur p olynomial in the v ariables t j a and ¯ σ ( j ) b is also an elementary Sc h ur p olynomial but now in the v ariables − ¯ t j,b . T o pro v e (109) we ﬁrst discuss the action of Λ 1 ,a and Λ > 2 ,b on g explicitly . Recalling (12) it is straightforw ard to see that (Λ 1 ,a g Λ > 2 ,b ) i,j = Z x k 1 ( i )+1 w 1 ,a 1 ( i ) ( x ) w 2 ,a 2 ( j ) ( x ) x k 2 ( j )+1 δ a 1 ( i ) ,a δ a 2 ( j ) ,b d µ ( x ) , and consequently the following expression holds ( W 0 g ¯ W − 1 0 ) i,j = p 1 X a =1 p 2 X b =1 Z x k 1 ( i )  X l ≥ 0 σ ( a ) l x l  w 1 ,a 1 ( i ) ( x ) w 2 ,a 2 ( j ) ( x )  X m ≥ 0 ¯ σ ( b ) m x m  x k 2 ( j ) δ a 1 ( i ) ,a δ a 2 ( j ) ,b d µ ( x ) , that leads directly to (109). 30 That the sign deﬁnition of the weigh ts is preserv ed under deformations is ensured b y the fact that all times t are real. Let us comment that these deformations could be also b e considered as ev olutions, and from hereon w e indistinctly talk ab out deformation/evolution. If the initial measures hav e b ounded supp ort then there is no problem with the exp onential behavior at ∞ of the E factors; ho wev er, for unbounded situations a discussion is needed for eac h case. The Gauss–Borel factorization problem g ~ n 1 ,~ n 2 ( t ) = S ( t ) − 1 ¯ S ( t ) , S ( t ) ∈ G − , ¯ S ( t ) ∈ G + , (110) with S ( t ) low er triangular and ¯ S ( t ) upp er triangular, will giv e the connection with integrable systems of T o da type. Let us assume that the w eights in ( ~ w 1 , ~ w 2 ) are of the form (28) and that conform an M-Nikishin, then Theo- rem 1 indicates that for small v alues of the times the new weigh ts are also in the M-Nikishin class, ensuring that ( ~ w 1 ( t ) , ~ w 2 ( t ) , µ ) is a p erfect system and therefore the Gauss–Borel factorization mak es sense. 3.2 Lax equations and the integrable hierarch y Let us introduce the Lax machinery associated with the Gauss–Borel factorization that will lead to a multi-component 2D T o da lattice hierarch y as describ ed in [28]: Deﬁnition 19. Asso ciate d with the deforme d Gauss–Bor el factorization we c onsider 1. Wave semi-inﬁnite matric es W ( t ) := S ( t ) W 0 ( t ) , ¯ W ( t ) := ¯ S ( t ) ¯ W 0 ( t ) . (111) 2. Wave Ψ a ( z , t ) := W ( t ) χ 1 ,a ( z ) , ¯ Ψ b ( z , t ) := ¯ W ( t ) χ ∗ 2 ,b ( z ) , (112) and adjoint wave semi-inﬁnite ve ctor functions 2 Ψ ∗ a ( z , t ) := ( W ( t ) − 1 ) > χ ∗ 1 ,a ( z ) , ¯ Ψ ∗ b ( z , t ) := ( ¯ W ( t ) − 1 ) > χ 2 ,b ( z ) . (113) 3. L ax semi-inﬁnite matric es L a ( t ) := S ( t )Λ 1 ,a S ( t ) − 1 , ¯ L b ( t ) := ¯ S ( t )Λ > 2 ,b ¯ S ( t ) − 1 . (114) 4. Zakhar ov–Shab at semi-inﬁnite matric es B j,a := ( L j a ) + , ¯ B j,b := ( ¯ L j b ) − , (115) wher e the subindex + indic ates the pr oje ction in the upp er triangular matric es while the subindex − the pr oje ction in the strictly lower triangular matric es. Observ e that L a Ψ a 0 = δ a,a 0 z Ψ a 0 , ¯ L > b ¯ Ψ ∗ b 0 = δ b,b 0 z − 1 ¯ Ψ ∗ b 0 . (116) W e also mention that the matrices S and ¯ S corresp ond to the Sato op erators (also kno wn as gauge op erators) of the in tegrable hierarc hy w e are deling with. Some times [9] the op erators L a are referred as resolv ents and the Lax name is reserved only for a conv enient linear combination of the resolven ts. The reader should notice that as S ( t ) ∈ G − and W 0 ( t ) ∈ G + the pro duct W ( t ) = S ( t ) W 0 ( t ) is well deﬁned as its co eﬃcien ts are ﬁnite sums instead of series, for ( ¯ W ( t ) − 1 ) > = ( ¯ S ( t ) − 1 ) > ( ¯ W 0 ( t )) − 1 ) > w e can apply the previous argumen t and therefore the pro duct is well deﬁned. How ev er, ( W ( t ) − 1 ) > = ( S ( t ) − 1 ) > ( W 0 ( t ) − 1 ) > is a pro duct of elemen ts whic h in volv es series instead of ﬁnite sum and its existence is not in principle ensured. The situation is 2 In this p oin t the reader should notice that there are tw o diﬀerences b et ween this deﬁnition of wa ve functions (also known as Bak er– Akheizer functions) and the one common in the literature, see for example [4]. Our mo diﬁcations are motiv ated by tw o facts, i) we prefer ¯ Ψ ∗ b to b e a p olynomial in z and not in z − 1 , up to plane-wav e factors, ii) we choose to hav e a direct connection b et ween wa ve functions and Cauch y transforms of p olynomials, with no z − 1 factors multiplying the Cauch y transforms when identiﬁed with wav e functions. If we denote by small ψ the w av e functions corresp onding to the scheme of for example [4] then we should hav e the following corresp ondence Ψ ( l ) a ( z ) ↔ ψ ( l ) a ( z ), z (Ψ ∗ a ) ( l ) ( z ) ↔ ( ψ ∗ a ) ( l ) ( z ), z − 1 ¯ Ψ ( l ) b ( z − 1 ) ↔ ¯ ψ ( l ) b ( z ) and ( ¯ Ψ ∗ b ) ( l ) ( z − 1 ) ↔ ( ¯ ψ ∗ b ) ( l ) ( z ). 31 repro duced with ¯ W ( t ) = ¯ S ( t ) ¯ W 0 ( t ), and the existence of the pro duct is not guaranteed. How ever, we notice that the sim ultaneous consideration of the factorization problems (110) and (33) leads S ( t ) − 1 ¯ S ( t ) = W 0 ( t ) S − 1 ¯ S ¯ W 0 ( t ) − 1 that shows tw o pro ducts inv olving series, namely W 0 ( t ) S − 1 and ¯ S ¯ W 0 ( t ) − 1 , but they are well deﬁned if w e assume the existence of b oth LU factorizations. F rom hereon we give for granted the existence of ¯ W and W − 1 , and as w e will see they indeed inv olve series, whic h in the con vergen t situation lead to Cauch y transforms. Prop osition 23. F or the wave functions we have Ψ ( k ) a ( z , t ) = A ( k ) a ( z , t ) E a ( z , t ) , ( ¯ Ψ ∗ b ) ( k ) ( z , t ) = ¯ A ( k ) b ( z , t ) ¯ E b ( z , t ) − 1 , (117) wher e A ( k ) a ( x, t ) , ¯ A ( k ) b ( x, t ) ar e the multiple ortho gonal p olynomials and dual p olynomials (in the x variable) c orr esp ond- ing to (109) . The evolve d line ar forms, asso ciate d with weights (109) , ar e Q ( k ) ( x, t ) := p 1 X a =1 A ( k ) a ( x, t ) w 1 ,a ( x, t ) = p 1 X a =1 Ψ ( k ) a ( x, t ) w 1 ,a ( x ) , (118) ¯ Q ( k ) ( x, t ) := p 2 X b =1 ( ¯ A ∗ b ) ( k ) ( x, t ) w 2 ,b ( x, t ) = p 2 X b =1 ( ¯ Ψ ∗ b ) ( k ) ( x, t ) w 2 ,b ( x ) , (119) which ar e bi-ortho gonal p olynomials of mixe d typ e for e ach t Z Q ( l ) ( t, x ) ¯ Q ( k ) ( t, x )d µ ( x ) = δ l,k , l, k ≥ 0 , (120) and ¯ Ψ ( k ) b ( z , t ) = Z Q ( k ) ( x, t ) z − x w 2 ,b ( x )d µ ( x ) , (Ψ ∗ a ) ( k ) ( z , t ) = Z ¯ Q ( k ) ( x, t ) z − x w 1 ,a ( x )d µ ( x ) . (121) Pr o of. F rom the deﬁnitions (112) and (113), and the factorization problem W g = ¯ W we conclude ¯ Ψ b = ¯ W χ ∗ 2 ,b = S ( W 0 g ) χ ∗ 2 ,b , Ψ ∗ a = ( W − 1 ) > χ ∗ 1 ,a = ( ¯ S − 1 ) > ( g ¯ W − 1 0 ) > χ ∗ 1 ,a . (122) W e get, in terms of the linear forms, the following iden tities ¯ Ψ ( k ) b ( z , t ) = Z Q ( k ) ( x, t ) z − x w 2 ,b ( x )d µ ( x ) , (Ψ ∗ a ) ( k ) ( z , t ) = Z ¯ Q ( k ) ( x, t ) z − x w 1 ,a ( x )d µ ( x ) , where the Cauch y transforms are understo od as b efore. 3 W e m ust stress in this point that these functions are not the evolv ed second kind functions of the linear forms ¯ C ( k ) b ( z , t ) := Z Q ( k ) ( x, t ) z − x w 2 ,b ( x, t )d µ ( x ) , ( C a ) ( k ) ( z , t ) := Z ¯ Q ( k ) ( x, t ) z − x w 1 ,a ( x, t )d µ ( x ) . (123) Theorem 6. F or j, j 0 = 1 , 2 , . . . , a, a 0 = 1 , . . . , p 1 and b, b 0 = 1 , . . . , p 2 the fol lowing diﬀer ential r elations hold 3 The reader should notice that there is a diﬀerence in this semi-inﬁnite context, appropriate for the construction of multiple orthogonal polynomials, and the bi-inﬁnite case which is the one considered in [37]. In the present context we do not hav e expressions, as we do hav e in the bi-inﬁnite situation, of the form ¯ Ψ ( k ) b ( z , t ) = ( P 0 + P 1 z − 1 + · · · ) exp  X j > 0 ¯ t j,b z j  , (Ψ ∗ a ) ( k ) ( z , t ) = ( Q 0 + Q 1 z − 1 + · · · ) exp  − X j > 0 t j,a z j  . The reason for this issue is ro oted into non-inv ertibility of Λ a . Indeed, for the semi-inﬁnite case, we hav e (Λ > a ) j χ ∗ a =  z j χ ∗ a  − = ⇒ exp  ∞ X j =1 c j (Λ > a ) j  χ ∗ a = [exp  ∞ X j =1 c j z j  χ ∗ a ] − where the subindex − stands for the negative p ow ers in z in the Lauren t expansion; while in the bi-inﬁnite case we drop the − subindex in the previous formulae. 32 1. Auxiliary line ar systems for the wave matric es ∂ W ∂ t j,a = B j,a W , ∂ W ∂ ¯ t j,b = ¯ B j,b W , ∂ ¯ W ∂ t j,a = B j,a ¯ W , ∂ ¯ W ∂ ¯ t j,b = ¯ B j,b ¯ W . (124) 2. Line ar systems for the wave and adjoint wave semi-inﬁnite matric es ∂ Ψ a 0 ∂ t j,a = B j,a Ψ a 0 , ∂ Ψ a 0 ∂ ¯ t j,b = ¯ B j,b Ψ a 0 , ∂ ¯ Ψ b 0 ∂ t j,a = B j,a ¯ Ψ b 0 , ∂ ¯ Ψ b 0 ∂ ¯ t j,b = ¯ B j,b ¯ Ψ b 0 , (125) ∂ Ψ ∗ a 0 ∂ t j,a = − B > j,a Ψ ∗ a 0 , ∂ Ψ ∗ a 0 ∂ ¯ t j,b = − ¯ B > j,b Ψ ∗ a 0 , ∂ ¯ Ψ ∗ b 0 ∂ t j,a = − B > j,a ¯ Ψ ∗ b 0 , ∂ ¯ Ψ ∗ b 0 ∂ ¯ t j,b = − ¯ B > j,b ¯ Ψ ∗ b 0 . (126) 3. Line ar systems for multiple ortho gonal p olynomials and their duals ∂ A a 0 ∂ t j,a = ( B j,a − δ a,a 0 x j ) A a 0 , ∂ A a 0 ∂ ¯ t j,b = ( ¯ B j,b ) A a 0 , ∂ ¯ A b 0 ∂ t j,a = − B > j,a ¯ A b 0 , ∂ ¯ A b 0 ∂ ¯ t j,b = ( − ¯ B > j,b + δ b,b 0 x j ) ¯ A b 0 . (127) 4. L ax e quations ∂ L a 0 ∂ t j,a = [ B j,a , L a 0 ] , ∂ L a 0 ∂ ¯ t j,b = [ ¯ B j,b , L a 0 ] , ∂ ¯ L b 0 ∂ t j,a = [ B j,a , ¯ L b 0 ] , ∂ ¯ L b 0 ∂ ¯ t j,b = [ ¯ B j,b , ¯ L b 0 ] . (128) 5. Zakhar ov–Shab at e quations ∂ B j,a ∂ t j 0 ,a 0 − ∂ B j 0 ,a 0 ∂ t j,a + [ B j,a , B j 0 ,a 0 ] = 0 , (129) ∂ ¯ B j,b ∂ ¯ t j 0 ,b 0 − ∂ ¯ B j 0 ,b 0 ∂ ¯ t j,b + [ ¯ B j,b , ¯ B j 0 ,b 0 ] = 0 , (130) ∂ B j,a ∂ ¯ t j 0 ,b 0 − ∂ ¯ B j 0 ,b 0 ∂ t j,a + [ B j,a , ¯ B j 0 ,b 0 ] = 0 . (131) Pr o of. T o pro ve (124) we pro ceed as follows. In the ﬁrst place we compute ∂ W 0 ∂ t j,a = Λ j 1 ,a W 0 , ∂ ¯ W 0 ∂ ¯ t j,b = (Λ > 2 ,b ) j ¯ W 0 , and in the second place we observe that ∂ W ∂ t j,a =  ∂ S ∂ t j,a S − 1 + L j a  W , ∂ ¯ W ∂ t j,a =  ∂ ¯ S ∂ t j,a S − 1  ¯ W , (132) ∂ W ∂ ¯ t j,b =  ∂ S ∂ ¯ t j,b S − 1  W , ∂ ¯ W ∂ ¯ t j,b =  ∂ ¯ S ∂ ¯ t j,b ¯ S − 1 + ¯ L j b  ¯ W . (133) No w, using the factorization problem we get ∂ S ∂ t j,a S − 1 + L j a = ∂ ¯ S ∂ t j,a ¯ S − 1 , ∂ ¯ S ∂ ¯ t j,b ¯ S − 1 + ¯ L j b = ∂ S ∂ ¯ t j,b S − 1 , whic h, taking the + part (upp er triangular) and the − part (strictly low er triangular) imply ∂ S ∂ t j,a S − 1 = − ( L j a ) − , ∂ ¯ S ∂ t j,a ¯ S − 1 = ( L j a ) + , ∂ ¯ S ∂ ¯ t j,b ¯ S − 1 = − ( ¯ L j b ) + , ∂ S ∂ ¯ t j,b S − 1 = ( ¯ L j b ) − , (134) so using (134) into (132) and (133) with the deﬁnitions (115) we obtain (124). The linear system (125) is obtained by inserting (112) into (124). 33 T o obtain the Lax equations (128) we take deriv atives of (114) ∂ L a 0 ∂ t j,a = h ∂ S ∂ t j,a S − 1 , L a 0 i = [ B j,a , L a 0 ] , ∂ ¯ L b 0 ∂ t j,a = h ∂ ¯ S ∂ t j,a ¯ S − 1 , ¯ L b 0 i = [ B j,a , ¯ L b 0 ] , ∂ L a 0 ∂ ¯ t j,b = h ∂ S ∂ ¯ t j,b S − 1 , L a 0 i = [ ¯ B j,b , L a 0 ] , ∂ ¯ L b 0 ∂ ¯ t j,b = h ∂ ¯ S ∂ ¯ t j,b ¯ S − 1 , ¯ L b 0 i = [ ¯ B j,b , ¯ L b 0 ] . Finally , (129) are obtained as compatibility conditions for (124). All these equations pro vide us with diﬀeren t descriptions of a m ulti-comp onen t integrable hierarch y of the 2D T o da lattice hierarc h y t yp e that rules the ﬂows of the multiple orthogonal p olynomials with resp ect to deformed weigh ts. This integrable hierarch y is the T o da t yp e extension of the multi-component KP hierarch y considered in [9]. 3.3 Darb oux–Miw a discrete ﬂo ws W e complete the previously considered contin uous ﬂows with discrete ﬂows, which we in tro duce through an iterated application of Darb oux transformations [5]. Deﬁnition 20. Given se quenc es of c omplex numb ers λ a := { λ a ( n ) } n ∈ Z ⊂ C , a = 1 , . . . , p 1 , ¯ λ b := { ¯ λ b ( n ) } n ∈ Z ⊂ C , b = 1 , . . . , p 2 , (135) (wher e ¯ λ is not intende d to denote the c omplex c onjugate of λ ) and two ve ctors, ( s 1 , . . . , s p 1 ) ∈ Z p 1 and ( ¯ s 1 , . . . , ¯ s p 2 ) ∈ Z p 2 , we c onstruct the fol lowing semi-inﬁnite matric es D 0 := p 1 X a =1 D 0 ,a , D 0 ,a :=      Q s a n =1 (Λ 1 ,a − λ a ( n )Π 1 ,a ) , s a > 0 , Π 1 ,a , s a = 0 , Q | s a | n =1 (Λ 1 ,a − λ a ( − n )Π 1 ,a ) − 1 , s a < 0 , ¯ D − 1 0 := p 2 X b =1  ¯ D − 1 0  b  ¯ D − 1 0  b :=      Q ¯ s b n =1 (Λ > 2 ,b − ¯ λ b ( n )Π 2 ,b ) , ¯ s b > 0 , Π 2 ,b , ¯ s b = 0 , Q | ¯ s b | n =1 (Λ > 2 ,b − ¯ λ b ( − n )Π 2 ,b ) − 1 , ¯ s b < 0 , (136) wher e s := { s a , ¯ s b } a =1 ,...,p 1 b =1 ,...,p 2 denotes the set of discr ete times, in terms of which we deﬁne the deforme d moment matrix g ( s ) = D 0 ( s ) g ¯ D 0 ( s ) − 1 . (137) Prop osition 24. The moment matrix g ( s ) has the same form as the moment matrix g but with new weights w 1 ,a ( s, x ) = D a ( x, s a ) w 1 ,a ( x ) , D a :=      Q s a n =1 ( x − λ a ( n )) , s a > 0 , 1 , s a = 0 , Q | s a | n =1 ( x − λ a ( − n )) − 1 , s a < 0 , w 2 ,b ( s, x ) = ¯ D b ( x, ¯ s b ) − 1 w 2 ,b ( x ) , ¯ D − 1 b :=      Q ¯ s b n =1 ( x − ¯ λ b ( n )) , ¯ s b > 0 , 1 , ¯ s b = 0 , Q | ¯ s b | n =1 ( x − ¯ λ b ( − n )) − 1 , ¯ s b < 0 , (138) Th us, the prop osed discrete evolution introduces new zero es and p oles in the w eigh ts at the p oints deﬁned by sequences of λ ’s. F or example, in the a -th direction, the s a ﬂo w in the positive direction, s a → s a + 1, in tro duces a new zero at the p oint λ a ( s a + 1), while if we mov e in the negative direction, s a → s a − 1, it introduces a simple p ole at λ a ( s a − 1). Let us stress that for the time b eing w e hav e not ensured the reality and positiveness/negativ eness of the evolv ed weigh ts, this will b e considered later on. 3.3.1 Miw a transformations Here we show that the discrete ﬂows just introduced can b e repro duced with the aid of Miwa shifts in the con tinuous v ariables. Deﬁnition 21. We c onsider two typ es of Miwa tr ansformations: 34 1. We intr o duc e the fol lowing time shifts t → t ∓ [ z − 1 ] a := n t j,a 0 ∓ δ a 0 ,a 1 j z j , ¯ t j,b 0 o j =1 , 2 ,..., a 0 =1 ,...,p 1 , b 0 =1 ,...,p 2 , (139) 2. Dual time shifts ar e t → t ± [ z − 1 ] b := n t j,a 0 , ¯ t j,b 0 ± δ b 0 ,b 1 j z j o j =1 , 2 ,..., a 0 =1 ,...,p 1 , b 0 =1 ,...,p 2 . (140) Prop osition 25. The Miwa tr ansformations pr o duc e the fol lowing eﬀe ct on the weights w 1 ,a 0 ( x, t ∓ [ z − 1 ] a , s ) =  1 − x z  ± δ a,a 0 w 1 ,a 0 ( x, t, s ) , w 2 ,b 0 ( x, t ∓ [ z − 1 ] a , s ) = w 2 ,b 0 ( x, t, s ) , (141) w 1 ,a 0 ( x, t ± [ z − 1 ] b , s ) = w 1 ,a 0 ( x, t, s ) , w 2 ,b 0 ( x, t ± [ z − 1 ] b , s ) =  1 − x z  ± δ b,b 0 w 2 ,b 0 ( x, t, s ) . (142) Pr o of. When we consider what happens to the evolutionary factors under these shifts we ﬁnd exp  X j t j,a 0 x j  → exp  X j  t j,a 0 ∓ δ a 0 ,a x j j z j  =  1 − x z  ∓ δ a 0 ,a exp  X j t j,a 0 x j  , (143) and therefore the weigh ts transform according to w 1 ,a 0 ( x, t, s ) →  1 − x z  ± δ a,a 0 w 1 ,a 0 ( x, t, s ) , (144) whic h is like the Darb oux transformations considered previously . F or the dual Miw a shifts w e consider what happ ens to the evolutionary factors under these shifts exp  − X j,b 0 ¯ t j,b 0 x j  → exp  − X j,b 0  ¯ t j,b 0 ± δ b 0 ,b x j j z j  =  1 − x z  ± 1 exp  − X j,b 0 ¯ t j,b 0 x j  , (145) and the transformation for the weigh ts is w 2 ,b 0 ( x, t, s ) →  1 − x z  ± δ b,b 0 w 2 ,a 0 ( x, t, s ) . (146) Th us, a comparison of (138), (141) and (142) leads to Prop osition 26. Miwa tr ansformations and discr ete ﬂows c an b e identiﬁe d as fol lows c a w 1 ,a ( x, t, s a ) =      w 1 ,a ( x, t − P s a n =1  λ a ( n ) − 1  a ) , s a > 0 , w 1 ,a ( x, t ) , s a = 0 , w 1 ,a ( x, t + P | s a | n =1  λ a ( − n ) − 1  a ) , s a < 0 , c a :=      Q s a n =1 ( − λ a ( n )) − 1 , s a > 0 1 , s a = 0 Q | s a | n =1 ( − λ a ( − n )) , s a < 0 , ¯ c b w 2 ,b ( x, t, ¯ s b ) =      w 2 ,b ( x, t + P ¯ s b n =1  ¯ λ b ( n ) − 1  b , x ) , ¯ s b > 0 , w 2 ,b ( x, t ) , ¯ s b = 0 , w 2 ,b ( x, t − P | ¯ s b | n =1  ¯ λ b ( − n ) − 1  b ) , ¯ s b < 0 , ¯ c b :=      Q ¯ s b n =1 ( − ¯ λ b ( n )) − 1 , ¯ s b > 0 1 , ¯ s b = 0 Q | ¯ s b | n =1 ( − ¯ λ b ( − n )) , ¯ s b < 0 . (147) As a conclusion, the discrete ﬂows and Miw a shifts in the contin uous ﬂows are the very same thing, and therefore w e could work with contin uous ﬂows and Miwa transformations or with contin uous/discrete ﬂows. This discussion justiﬁes the Miwa part in the name w e ga ve to these discrete ﬂows. 35 3.3.2 Bounded from below measures Of course, in order to preserve the link with multiple orthogonal polynomials, these discrete ﬂows m ust preserve the realit y , regularity and sign constance of the weigh ts, which generically is not the case. When the support of the w eights is b ounded from b elo w, i.e. there are ﬁnite real n umbers K a and K b , suc h that supp( w 1 ,a d µ ) ⊂ [ K a , ∞ ) and supp( w 2 ,b d µ ) ⊂ [ ¯ K b , ∞ ), a possible solution is to place all the new zero es and p oles in the real line but outside the corresp onding supp ort, λ a ( n ) < inf(supp( w 1 ,a d µ )) and ¯ λ b ( n ) < inf(supp( w 2 ,b d µ )). A diﬀeren t approac h, which will b e considered in App endix B, is to arrange the zero es in complex conjugate pairs. T o analyze the consequence of the discrete ﬂows on the integrable hierarch y we introduce tw o sets of shifts op erators: Deﬁnition 22. 1. L et us c onsider the sets of shift op er ators { T a } p 1 a =1 and { ¯ T b } p 2 b =1 , wher e T a stands for the shift s a 7→ s a + 1 and ¯ T b stands for s b 7→ ¯ s b + 1 . The r est of the variables { s a 0 , ¯ s b 0 } wil l r emain c onstant. 2. We intr o duc e q a := I − Π 1 ,a ( I + λ a ( s a + 1)) + Λ 1 ,a , ¯ q b := I − Π 2 ,b ( I + ¯ λ b ( ¯ s b + 1)) + Λ > 2 ,b . (148) 3. We also deﬁne the op er ators δ a := S q a S − 1 = I − C a ( I + λ a ( s a + 1)) + L a , ¯ δ b := ¯ S ¯ q b ¯ S − 1 = I − ¯ C b ( I + ¯ λ b ( ¯ s b + 1)) + ¯ L b , (149) C a := S Π 1 ,a S − 1 , ¯ C b := ¯ S Π 2 ,b ¯ S − 1 . Her e the matric es δ a and ¯ δ b ar e c al le d lattic e r esolvents. 4. Final ly the semi-inﬁnite wave matric es W := S D 0 , ¯ W := ¯ S ¯ D 0 . (150) Observ e that ( T a D 0 ) D − 1 0 = q a , ¯ D − 1 0 ( T a ¯ D 0 ) = I , ¯ D 0 ( ¯ T b ¯ D − 1 0 ) = ¯ q b , ( ¯ T b D 0 ) D − 1 0 = I . (151) When we assume that the semi-inﬁnite matrices δ a and ¯ δ b are LU factorizable as in (33), i.e. all their principal minors do not v anish, we can write δ a = δ − 1 a, − δ a, + , ¯ δ b = ¯ δ − 1 b, − ¯ δ b, + , (152) where δ a, − and ¯ δ b, − are lo wer matrices as is S in (33), and δ a, + and ¯ δ b, + are upp er matrices as ¯ S in (33). W e no w show that when the deformed moment matrix g ( s ) is factorizable, and therefore the multiple orthogonality mak es sense, the follo wing holds Prop osition 27. If the deforme d moment matrix g ( s ) is factorizable for al l values of s then so is δ a and ¯ δ b with δ a, + = ( T a ¯ S ) ¯ S − 1 , δ a, − = ( T a S ) S − 1 , ¯ δ b, + = ( ¯ T b ¯ S ) ¯ S − 1 , ¯ δ b, + = ( ¯ T b S ) S − 1 . (153) Pr o of. When we apply the discrete shifts to the Gauss–Borel factorization problem g ( s ) = S − 1 ( s ) ¯ S ( s ) w e get T a ( S − 1 ) T a ( ¯ S ) = T a g ( s ) = ( T a D 0 ) D − 1 0 g ( s ) = q a g ( s ) ⇒ (( T a S ) S − 1 ) − 1 ( T a ¯ S ) ¯ S − 1 = δ a , ¯ T b ( S − 1 ) ¯ T b ( ¯ S ) = ¯ T b g ( s ) = g ( s ) ¯ D 0 ( T b ¯ D − 1 0 ) = g ( s ) ¯ q b ⇒ (( ¯ T b S ) S − 1 ) − 1 (( ¯ T b ¯ S ) ¯ S − 1 = ¯ δ b , and the desired result follows. Therefore, we can consider the following Deﬁnition 23. The semi-inﬁnite matric es ω a and ¯ ω b ar e given by ω a := δ a, − δ a = δ a, + , ¯ ω b := ¯ δ b, − = ¯ δ b, + ¯ δ − 1 b , (154) 36 and show that Prop osition 28. 1. The fol lowing auxiliary line ar systems T a W = ω a W , T a ¯ W = ω a ¯ W , ¯ T b W = ¯ ω b W , ¯ T b ¯ W = ¯ ω b ¯ W , (155) ar e satisﬁe d. 2. The L ax matric es fulﬁl l the fol lowing r elations T a L a 0 = ω a L a 0 ω − 1 a , T a ¯ L b = ω a ¯ L b ω − 1 a , ¯ T b L a = ¯ ω b L a ¯ ω − 1 b , ¯ T b ¯ L b 0 = ¯ ω b ¯ L b 0 ¯ ω − 1 b . (156) 3. The fol lowing discr ete Zakhar ov–Shab at c omp atibility c onditions hold ( T a ω a 0 ) ω a = ( T a 0 ω a ) ω a 0 , ( T a ¯ ω b ) ω a = ( ¯ T b ω a ) ¯ ω b , ( ¯ T b ¯ ω b 0 ) ¯ ω b = ( ¯ T b 0 ¯ ω b ) ¯ ω b 0 . (157) 4. When the discr ete and c ontinous ﬂows ar e c onsider e d simultane ously, the fol lowing e quations T a 0 B j,a = ( ∂ j,a ω a 0 ) ω − 1 a 0 + ω a 0 B j,a ω − 1 a 0 , ¯ T b B j,a = ( ∂ j,a ¯ ω b ) ¯ ω − 1 b + ¯ ω b B j,a ¯ ω − 1 b , T a ¯ B j,b = ( ¯ ∂ j,b ω a ) ω − 1 a + ω a ¯ B j,b ω − 1 a , ¯ T b 0 ¯ B j,b = ( ¯ ∂ j,b ¯ ω b 0 ) ¯ ω − 1 b 0 + ¯ ω b 0 ¯ B j,b ¯ ω − 1 b 0 , (158) ar e obtaine d. Pr o of. W e compute T a W = ( T a S )( T a D 0 ) = ( T a S ) S − 1 S q a S − 1 S D 0 = δ a, − δ a W = δ a, + W , T a ¯ W = ( T a ¯ S ) ¯ D 0 = ( T a ¯ S ) ¯ S − 1 ¯ S ¯ D 0 = δ a, + ¯ W , ¯ T b W = ( ¯ T b S ) D 0 = ( ¯ T b S ) S − 1 S D 0 = ¯ δ b, − W , ¯ T b ¯ W = ( ¯ T b ¯ S )( T b ¯ D 0 ) = ( T b ¯ S ) ¯ S − 1 ¯ S ¯ q − 1 b ¯ S − 1 ¯ S ¯ D 0 = ¯ δ b, + ¯ δ − 1 b ¯ W = ¯ δ b, − ¯ W , from where we deduce (155), which in turn imply (156) and (157). The simultaneous consideration of contin uous and discrete ﬂows leads to the replacement W 0 → W 0 D 0 and ¯ W 0 → ¯ W 0 ¯ D 0 , and the corresponding mo diﬁcation of the weigh t’s ﬂo ws is ac hieved by the m ultiplication of the contin uous and discrete evolutionary factors, in this context we also ha ve (158). These discrete ﬂo ws could b e understo o d as a sequence of Darb oux transformations of LU and U L t yp es in the terminology of [5], whic h motiv ates the Darb oux part in name we give to these discrete ﬂows. In fact, we ha ve that the lattice resolv ents satisfy δ a = δ − 1 a, − δ a, + ⇒ T a δ a = ω a δ a ω − 1 a = δ a, + δ − 1 a, − δ a, + δ − 1 a, + = δ a, + δ − 1 a, − , ¯ δ b = ¯ δ − 1 b, − ¯ δ b, + ⇒ ¯ T b ¯ δ b = ¯ ω b ¯ δ b ¯ ω − 1 b = ¯ δ b, − ¯ δ − 1 b, − ¯ δ b, + ¯ δ − 1 b, − = ¯ δ b, + ¯ δ − 1 b, − , whic h amounts to the t ypical p erm utation of the LU factorization to the U L factorization. When there is only one comp onen t we hav e δ = L + λ and ¯ δ = ¯ λ + ¯ L and the shift corresp onds to the classical LU or U L Darboux transformations. If A ( k ) a ( x, s ) , ¯ A ( k ) b ( x, s ) are the multiple orthogonal p olynomials and dual p olynomials in the x v ariable corresp ond- ing to the discrete evolution of the w eights (138) resp ectiv ely we hav e the discrete v ersion of Prop osition 23 Prop osition 29. The wave and adjoint wave functions (150) ar e Ψ ( k ) a ( z , s ) = A ( k ) a ( z , s ) D a ( z , s a ) ( ¯ Ψ ∗ b ) ( k ) ( z , s ) = ¯ A ( k ) b ( z , s ) ¯ D b ( z , ¯ s b ) − 1 , (159) and the the line ar forms Q ( k ) ( x, s ) = p 1 X a =1 A ( k ) a ( x, s ) w 1 ,a ( x, s ) , ¯ Q ( k ) ( x, s ) = p 2 X b =1 ( ¯ A b ) ( k ) ( x, s ) w 2 ,b ( x, s ) , (160) 37 asso ciate d with the weights w 1 ,a ( x, s ) , w 2 ,b ( x, s ) , c an b e expr esse d as Q ( k ) ( x, s ) := p 1 X a =1 Ψ ( k ) a ( x, s ) w 1 ,a ( x ) , ¯ Q ( k ) ( x, s ) := p 2 X b =1 ( ¯ Ψ ∗ b ) ( k ) ( x, s ) w 2 ,b ( x ) , (161) in terms of which we have the e quations ¯ Ψ ( k ) b ( z , s ) = Z Q ( k ) ( x, s ) z − x w 2 ,b ( x )d µ ( x ) , (Ψ ∗ a ) ( k ) ( z , s ) = Z ¯ Q ( k ) ( x, s ) z − x w 1 ,a ( x )d µ ( x ) . (162) Here the Cauch y transforms must be interpreted in exactly the same terms as in Prop osition 8. Observe that (162) do not corresp ond to the functions of the second kind ¯ C ( k ) b ( z , s ) := Z Q ( k ) ( x, s ) z − x w 2 ,b ( x, s )d µ ( x ) , C ( k ) a ( z , s ) := Z ¯ Q ( k ) ( x, s ) z − x w 1 ,a ( x, s )d µ ( x ) . (163) Notice also that from (149)-(152), relations that hold true for any g and not only for the moment matrix, we get Lemma 6. We have that ω a = ω a, 0 Λ | ~ n 1 |− n 1 ,a +1 + ω a, 1 Λ | ~ n 1 |− n 1 ,a + · · · + ω a, | ~ n 1 |− n 1 ,a +1 , ¯ ω b = ¯ ω b, 0 (Λ > ) | ~ n 2 |− n 2 ,b +1 + ¯ ω b, 1 (Λ > ) | ~ n 2 |− n 2 ,b + · · · + ¯ ω b, | ~ n 2 |− n 2 ,b +1 , ω > a = ρ a, 0 (Λ > ) | ~ n 1 |− n 1 ,a +1 + ρ a, 1 (Λ > ) | ~ n 1 |− n 1 ,a + · · · + ρ a, | ~ n 1 |− n 1 ,a +1 , ¯ ω > b = ¯ ρ b, 0 Λ | ~ n 2 |− n 2 ,b +1 + ¯ ρ b, 1 Λ | ~ n 2 |− n 2 ,b + · · · + ρ b, | ~ n 2 |− n 2 ,b +1 , (164) for some diagonal semi-inﬁnite matric es ω a,j = diag( ω a,j (0) , ω a,j (1) , . . . ) , ¯ ω b,j = diag( ¯ ω b,j (0) , ¯ ω b,j (1) , . . . ) , ρ a,j = diag( ρ a,j (0) , ρ a,j (1) , . . . ) , ¯ ρ b,j = diag( ¯ ρ b,j (0) , ¯ ρ b,j (1) , . . . ) , (165) with ρ a,j ( k ) := ω a,j ( k − | ~ n 1 | + n 1 ,a − 1 + j ) , ¯ ρ b,j ( k ) := ¯ ω b,j ( k + | ~ n 2 | − n 2 ,b + 1 − j ) . (166) that with Deﬁnition 24. We deﬁne γ a,a 0 ( s, x ) := (1 − δ a,a 0 (1 + λ a ( s a + 1) − x )) , γ b,b 0 ( s, x ) := (1 − δ b,b 0 (1 + ¯ λ b ( ¯ s b + 1) − x )) , (167) leads to Prop osition 30. The fol lowing e quations ( T a 0 A ( k ) a ) γ a,a 0 = ω a 0 , 0 ( k ) A ( k + | ~ n 1 |− n 1 ,a 0 +1) a + · · · + ω a 0 , | ~ n 1 |− n 1 ,a 0 +1 ( k ) A ( k ) a , ¯ T b 0 A ( k ) a = ¯ ω b 0 , 0 ( k ) A ( k −| ~ n 2 | + n 2 ,b 0 − 1) a + · · · + ¯ ω b 0 , | ~ n 2 |− n 2 ,b 0 +1 ( k ) A ( k ) a . (168) ρ a 0 , 0 ( k )( T a 0 ¯ A ( k −| ~ n 1 | + n 1 ,a 0 − 1) b ) + · · · + ρ a 0 , | ~ n 1 |− n 1 ,a 0 +1 ( k )( T a 0 ¯ A ( k ) b ) = ¯ A ( k ) b ,  ¯ ρ b 0 , 0 ( k )( ¯ T b 0 ¯ A ( k + | ~ n 2 |− n 2 ,b 0 +1) b ) + · · · + ¯ ρ b 0 , | ~ n 2 |− n 2 ,b 0 +1 ( k )( ¯ T b 0 ¯ A ( k ) b )  ¯ γ b,b 0 = ¯ A ( k ) b , (169) ar e fulﬁl le d. Pr o of. F or (168) recall the discrete auxiliary systems for W , while for (169) just consider that ω > a T a (( ¯ W − 1 ) > ) = ( ¯ W − 1 ) > , ¯ ω > b ¯ T b (( ¯ W − 1 ) > ) = ( ¯ W − 1 ) > . Notice that relations (168) and (169) are among multiple orthogonal p olynomials in the same ladder but with diﬀeren t weigh ts, they link the p olynomials for the w eights w 1 ,a , w 2 ,b with those with T a 0 w 1 ,a , T a 0 w 2 ,b or ¯ T b 0 w 1 ,a , ¯ T b 0 w 2 ,b . 38 3.4 Symmetries, recursion relations and string equations W e now return to the discussion of the symmetry of the momen t matrix that w e started in § 2.6 but with evolv ed w eights and the use of Lax matrices. The ﬁrst observ ation is the follo wing Prop osition 31. The j -th p ower of the evolve d Jac obi typ e matrix intr o duc e d in § 2.6 is r elate d with L ax matric es thr ough what we c al l a string e quation: J j = p 1 X a =1 L j a = p 2 X b =1 ¯ L j b , j = 1 , 2 , . . . , (170) and the multiple ortho gonal p olynomials ar e eigen-ve ctors: J j A a 0 = x j A a 0 , ( J j ) > ¯ A b 0 = x j ¯ A b 0 , (171) for a 0 = 1 , . . . , p 1 and b 0 = 1 , . . . , p 2 . Pr o of. Using (68) it can b e prov en by induction on j that for any j ≥ 1 the following equation holds Λ j 1 ,a g Π 2 ,b = Π 1 ,a g (Λ > 2 ,b ) j , (172) so that L j a ¯ C b = C a ¯ L j b . (173) Summing ov er a, b we deduce (170). Moreo ver (171) is obtained as follo ws J j A a 0 = S p 1 X a =1 Λ j 1 ,a S − 1 S χ 1 ,a 0 = x j A a 0 , (174) ( J j ) > ¯ A a 0 = ( ¯ S − 1 ) > p 2 X b =1 Λ j 2 ,a ¯ S > ( ¯ S − 1 ) > χ 2 ,b 0 = x j ¯ A b 0 . (175) W e are ready to sho w that the symmetry (68) induces a corresp onding inv ariance on Lax matrices and multiple orthogonal p olynomials Prop osition 32. The fol lowing r elations hold for j = 1 , 2 , . . .  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  L a 0 = 0 ,  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  ¯ L b 0 = 0 , (176)  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  A a 0 = 0 ,  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  ¯ A b 0 = 0 . (177) Pr o of. See App endix A. 3.5 Bilinear equations and τ -functions The pro of of the bilinear iden tit y needs three lemmas. F or the ﬁrst one, let W ~ n 1 ,~ n 2 , ¯ W ~ n 1 ,~ n 2 b e the w a ve matrices asso ciated with the momen t matrix g ~ n 1 ,~ n 2 ; so that, W ~ n 1 ,~ n 2 g ~ n 1 ,~ n 2 = ¯ W ~ n 1 ,~ n 2 . Then, we ha ve Lemma 7. The wave matric es asso ciate d with diﬀer ent c omp ositions and times satisfy W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 1 ,~ n 1 W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 = ¯ W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 2 ,~ n 2 ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 , (178) Pr o of. W e consider simultaneously the following equations W ~ n 1 ,~ n 2 ( t, s ) g = ¯ W ~ n 1 ,~ n 2 ( t, s ) , W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) π ~ n 0 1 ,~ n 1 g π > ~ n 0 2 ,~ n 2 = ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) , where g = g ~ n 1 ,~ n 2 , and we get W ~ n 1 ,~ n 2 ( t, s ) − 1 ¯ W ~ n 1 ,~ n 2 ( t, s ) = π > ~ n 0 1 ,~ n 1 W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) π ~ n 0 2 ,~ n 2 = g , and the result b ecomes evident. 39 F or the second one, let ( · ) − 1 denote the co eﬃcien t in z − 1 in the Laurent expansi´ on around z = ∞ (place where the Cauch y transforms make sense). Lemma 8. F or the ve ctors χ a the fol lowing formulae hold  p X a =1 χ a ( χ ∗ a ) >  − 1 =  p X a =1 χ ∗ a χ > a  − 1 = I , and therefore Lemma 9. F or any c ouple of semi-inﬁnite matric es U and V we have U V =  p 1 X a =1 ( U χ 1 ,a )  V > χ ∗ 1 ,a  >  − 1 (179) =  p 2 X b =1 ( U χ ∗ 2 ,b )  V > χ 2 ,b  >  − 1 , (180) Pr o of. It follows easily from Lemma 8:  p 1 X a =1 ( U χ 1 ,a )  V > χ ∗ 1 ,a  >  − 1 = U  p 1 X a =1 χ 1 ,a ( χ ∗ 1 ,a ) >  − 1 V = U V ,  p 2 X b =1 ( U χ ∗ 2 ,b )  V > χ 2 ,b  >  − 1 = U  p 2 X b =1 χ ∗ 2 ,b χ > 2 ,b  − 1 V = U V . W e ha ve the following Theorem 7. 1. The wave functions and their c omp anions satisfy p 1 X a =1 I ∞ Ψ ( k ) ~ n 1 ,~ n 2 ,a ( z , t, s )(Ψ ∗ ~ n 0 1 ,~ n 0 2 ,a ) ( l ) ( z , t 0 , s 0 )d z = p 2 X b =1 I ∞ ¯ Ψ ( k ) ~ n 1 ,~ n 2 ,b ( z , t, s )( ¯ Ψ ∗ ~ n 0 1 ,~ n 0 2 ,b ) ( l ) ( z , t 0 , s 0 )d z . 2. Multiple ortho gonal p olynomials, their duals and the c orr esp onding se c ond kind functions ar e linke d by p 1 X a =1 I ∞ A ( k ) ~ n 1 ,~ n 2 ,a ( z , t, s ) ¯ C ( l ) ~ n 0 1 ,~ n 0 2 ,a ( z , t 0 , s 0 ) E a ( z )d z = p 2 X b =1 I ∞ C ( k ) ~ n 1 ,~ n 2 ,b ( z , t, s ) ¯ A ( l ) ~ n 0 1 ,~ n 0 2 ,b ( z , t 0 , s 0 ) ¯ E b ( z )d z , (181) wher e E a := ( E a D a )( z , t, s )(( E a D a )( z , t 0 , s 0 )) − 1 , ¯ E b := ( ¯ E b ¯ D b )( z , t, s )(( ¯ E b ¯ D b )( z , t 0 , s 0 )) − 1 . Pr o of. 1. If we set in (179) U = W ~ n 1 ,~ n 2 ( t, s ) and V = π > ~ n 0 1 ,~ n 1 W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 and in (180) w e put U = ¯ W ~ n 1 ,~ n 2 ( t, s ) and V = π > ~ n 0 2 ,~ n 2 ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 attending to (178), recalling that Ψ ~ n 1 ,~ n 2 ,a = W ~ n 1 ,~ n 2 χ ~ n 1 ,a , ¯ Ψ ~ n 1 ,~ n 2 ,b = ¯ W ~ n 1 ,~ n 2 χ ∗ ~ n 2 ,b and observing that Ψ ∗ ~ n 0 1 ,~ n 0 2 ,a = ( W − 1 ~ n 0 1 ,~ n 0 2 ) > π ~ n 0 1 ,~ n 1 χ ∗ ~ n 1 ,a and ¯ Ψ ∗ ~ n 0 1 ,~ n 0 2 ,b = ( ¯ W − 1 ~ n 0 1 ,~ n 0 2 ) > π ~ n 0 2 ,~ n 2 χ ~ n 2 ,b w e get the stated bilinear equation for the wa ve functions. 4 2. W e can write W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 1 ,~ n 1 W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 = ( S ~ n 1 ,~ n 2 ( t, s ) W 0 ,~ n 1 ( t, s ) π > ~ n 0 1 ,~ n 1 ( W 0 ,~ n 0 1 ( t 0 , s 0 )) − 1 π ~ n 0 1 ,~ n 1 ) π > ~ n 0 1 ,~ n 1 S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 , whic h strongly suggests to consider in (179) U = S ~ n 1 ,~ n 2 ( t, s ) W 0 ,~ n 1 ( t, s ) π > ~ n 0 1 ,~ n 1 ( W 0 ,~ n 0 1 ( t 0 , s 0 )) − 1 π ~ n 0 1 ,~ n 1 , V = π > ~ n 0 1 ,~ n 1 S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 . 4 The reader familiarized with T o da bilinear equations should notice that in the right hand term we are working at z = ∞ instead of, as customary , at z = 0; the reason is that for the deﬁnition of ¯ A b we hav e used χ 2 instead of χ ∗ 2 , in order to get polynomials in z , while normally one gets p olynomials in z − 1 . See fo otnote 2. Moreo ver, due to the redeﬁnition of the wa ve functions there is no d z 2 π i z factor 40 Analogously ¯ W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 1 ,~ n 1 ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 = ( ¯ S ~ n 1 ,~ n 2 ( t, s ) ¯ W 0 ,~ n 1 ( t, s ) π > ~ n 0 1 ,~ n 1 ( ¯ W 0 ,~ n 0 1 ( t 0 , s 0 )) − 1 π ~ n 0 1 ,~ n 1 ) π > ~ n 0 1 ,~ n 1 ¯ S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 , suggest to set in (180) U = ¯ S ~ n 1 ,~ n 2 ( t, s ) ¯ W 0 ,~ n 2 ( t, s ) π > ~ n 0 2 ,~ n 2 ( ¯ W 0 ,~ n 0 2 ( t 0 , s 0 )) − 1 π ~ n 0 2 ,~ n 2 , V = π > ~ n 0 2 ,~ n 2 ¯ S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 ) − 1 . The application of (179),(180) and (178) gives the alternative bilinear relations (181) where we ha v e used the ev olved Cauch y transforms (123) and in tro duce the ev olutionary factors E a := ( E a D a )( z , t, s )(( E a D a )( z , t 0 , s 0 )) − 1 , ¯ E b := ( ¯ E b ¯ D b )( z , t, s )(( ¯ E b ¯ D b )( z , t 0 , s 0 )) − 1 . The factors inv olv ed in this deﬁnition were introduced in (109) and (138), so that we assume the discrete ﬂo ws within the b ounded from b elo w support scenario, while if we consider the tw o-step discrete ﬂo ws the replacement of the D -factors by the D 0 - factors (220) is required. It can be shown that for certain weigh ts, for which w e can use the F ubini and Cauc hy theorems, and when one only considers a ﬁnite num b er of contin uous ﬂows that the r.h.s and l.hs. in this bilinear relations are prop ortional to R R Q ( k ) ~ n 1 ,~ n 2 ( x, t ) ¯ Q ( l ) ~ n 0 1 ,~ n 0 2 ( x, t 0 )d µ ( x ). This is a direct consequence of Prop osition 33. We have the fol lowing identity Z R Q ~ n 1 ,~ n 2 ( x, t, s ) ¯ Q > ~ n 0 1 ,~ n 0 2 ( x, t 0 , s 0 )d µ ( x ) = W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 1 ,~ n 1 ( W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 (182) = ¯ W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 2 ,~ n 2 ( ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 . (183) Pr o of. See App endix A. No w, we will perform a full characterization of the τ -functions asso ciated with the m ultiple orthogonal p olynomials deﬁned in this pap er. Deﬁnition 25. L et us deﬁne the fol lowing matric es g [ l +1] + a : =        g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 ,l . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 ,l g l + a , 0 g l + a , 1 · · · g l + a ,l − 1 g l + a ,l        ¯ g [ l +1] + b : =        g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 , ¯ l + b g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 , ¯ l + b . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 , ¯ l + b g l, 0 g l, 1 · · · g l,l − 1 g l, ¯ l + b        . (184) The matrix g [ l +1] + a is obtained from g [ l +1] replacing the last row (op eration denoted by a dashed line) b y ( g l + a , 0 , g l + a , 1 , . . . , g l + a ,l − 1 , g l + a ,l ) , and ¯ g [ l +1] + b is obtained from g [ l +1] replacing the last column by ( g 0 , ¯ l + b , g 1 , ¯ l + b , . . . , g l − 1 , ¯ l + b , g l, ¯ l + b ) > . It is clear that if a 1 ( l ) = a then g [ l +1] + a = g [ l +1] and if a 2 ( l ) = b then ¯ g [ l +1] + b = g [ l +1] . The minors of the these matrices (184) will b e denoted as M [ l +1] i,j = ¯ M [ l +1] i,j for g [ l +1] , M [ l +1] + a,i,j for g [ l +1] + a and ¯ M [ l +1] + b,i,j for ¯ g [ l +1] + b . Now we in tro duce the following determinants that are cofactors of the previously deﬁned matrices Deﬁnition 26. The τ -functions ar e deﬁne d as fol lows τ ( l ) + a, − a 0 := ( − 1) l + l − a 0 M [ l +1] + a,l − a 0 ,l , τ ( l ) − b, − a := ( − 1) ¯ l − b + l − a M [ l +1] l − a , ¯ l − b , (185) ¯ τ ( l ) + b, − b 0 := ( − 1) l + ¯ l − b 0 ¯ M [ l +1] + b,l, ¯ l − b 0 , ¯ τ ( l ) − a, − b := ( − 1) l − a + ¯ l − b ¯ M [ l +1] l − a , ¯ l − b . (186) Mor e over, 41 1. If a 1 ( l ) = a then we denote τ ( l ) − a 0 := τ ( l ) + a, − a 0 and ¯ τ ( l ) − b := ¯ τ ( l ) − a, − b . 2. If a 2 ( l ) = b then we denote τ ( l ) − a := τ ( l ) − b, − a and ¯ τ ( l ) − b 0 := ¯ τ ( l ) + b, − b 0 . 3. We also intr o duc e τ ( l ) = ¯ τ ( l ) := det g [ l ] and τ ( l +1) + a := det g [ l +1] + a , ¯ τ ( l +1) + b := det ¯ g [ l +1] + b . If a 1 ( l ) = a then τ ( l +1) + a = τ ( l +1) , and if a 2 ( l ) = b then ¯ τ ( l +1) + b = τ ( l +1) . Giv en a p erfect combination ( µ, ~ w 1 , ~ w 2 ) and the corresp onding set of multiple orthogonal p olynomials { A [ ~ ν 1 ; ~ ν 2 ] ,a } p 1 a =1 , with degree vectors such that | ~ ν 1 | = | ~ ν 2 | + 1, there exists a ( ~ n 1 , ~ n 2 ) ladder and an integer l with | ~ ν 1 | = l + 1 and | ~ ν 2 | = l such that the p olynomials { A ( l ) a } p 1 a =1 coincide with { A [ ~ ν 1 ; ~ ν 2 ] ,a } p 1 a =1 . The ﬁnal result do es not dep end up on the particular ( ~ n 1 , ~ n 2 ) ladder we choose to get up to the given degrees in the ladder; ho w ever, the τ -functions do indeed dep end on the ladder chosen through a global sign. A simple sign-ﬁxing rule is to c ho ose the ladder ~ n 1 = ~ ν 1 and ~ n 2 = ~ ν 2 + ~ e p 2 . W e deﬁne τ [ ~ ν 1 ; ~ ν 2 ] := τ ( l ) ~ ν 1 ,~ ν 2 , l = | ~ ν 1 | − 1 = | ~ ν 2 | , and we deduce Prop osition 34. Given de gr e e ve ctors ( ~ ν 1 , ~ ν 2 ) such that | ~ ν 1 | = | ~ ν 2 | + 1 , a c omp osition with ~ n 1 = ~ ν 1 and ~ n 2 = ~ ν 2 + ~ e p 2 and l = | ~ ν 1 | − 1 = | ~ ν 2 | , we have the fol lowing identities τ ( l ) + a, − a 0 = ε 1 , 1 ( a, a 0 ) τ [ ~ ν 1 − ~ e 1 ,a 0 + ~ e 1 ,a ; ~ ν 2 ] , ¯ τ ( l ) + b, − b 0 = ε 2 , 2 ( b, b 0 ) τ [ ~ ν 1 ; ~ ν 2 − ~ e 2 ,b 0 + ~ e 2 ,b ] , τ ( l ) − b, − a = ¯ τ ( l ) − a, − b = ε 2 , 1 ( b, a ) τ [ ~ ν 1 + ~ e 1 ,p 1 − ~ e 1 ,a ; ~ ν 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] , wher e ε 1 , 1 ( a, a 0 ) := ( − 1) P a j =1 ν 1 ,j + P a 0 j =1 ν 1 ,j + δ a,p 1 − 1 , a 0 < a ε 1 , 1 ( a, a 0 ) := ( − 1) P a j =1 ν 1 ,j + P a 0 j =1 ν 1 ,j + δ a 0 ,p 1 , a 0 > a ε 2 , 2 ( b, b 0 ) := ( − 1) P b j =1 ν 2 ,j + P b 0 j =1 ν 2 ,j − 1 , b 0 < b ε 2 , 2 ( b, b 0 ) := ( − 1) P b j =1 ν 2 ,j + P b 0 j =1 ν 2 ,j , b 0 > b ε 2 , 1 ( b, a ) := ( − 1) P b j =1 ν 2 ,j + P a j =1 ν 1 ,j + δ b,p 2 , ε 1 , 1 ( a, a ) := 1 = ε 2 , 2 ( b, b ) . In p articular τ ( l ) − a = ε 1 , 1 ( p 1 , a ) τ [ ~ ν 1 + ~ e 1 ,p 1 − ~ e 1 ,a ; ~ ν 2 ] , ¯ τ ( l ) − b = ε 2 , 2 ( p 2 , b ) τ [ ~ ν 1 ; ~ ν 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] , τ ( l +1) + a = ε 1 , 1 ( a, p 1 ) τ [ ~ ν 1 + ~ e 1 ,a ; ~ ν 2 + ~ e 2 ,p 2 ] , ¯ τ ( l +1) + b = ε 2 , 2 ( b, p 2 ) τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,b ] . W e now pro ceed to give the τ -function represen tation of m ultiple orthogonal p olynomials, their duals, second kind functions and bilinear equations. The τ -functions allo w for compact expressions for the multiple orthogonal p olynomials: Prop osition 35. The mixe d multiple ortho gonal p olynomials A ( l ) a , A ( l ) + a 0 ,a and A ( l ) − b,a have the fol lowing τ -function r epr esentation A ( l ) a ( z ) = A (II ,a 1 ( l )) [ ~ ν 1 ( l ); ~ ν 2 ( l − 1)] ,a = z ν 1 ,a ( l ) − 1 τ ( l ) − a ( t − [ z − 1 ] a ) τ ( l ) ( t ) , l ≥ 1 , (187) A ( l ) + a 0 ,a ( z ) = A (II ,a 0 ) [ ~ ν 1 ( l − 1)+ ~ e 1 ,a 0 ; ~ ν 2 ( l − 1)] ,a = z ν 1 ,a ( l − 1)+ δ a,a 0 − 1 τ ( l ) + a 0 , − a ( t − [ z − 1 ] a ) τ ( l ) ( t ) , l ≥ 1 , (188) A ( l ) − b,a ( z ) = A (I ,b ) [ ~ ν 1 ( l ); ~ ν 2 ( l ) − ~ e 2 ,b ] ,a = z ν 1 ,a ( l ) − 1 τ ( l ) − b, − a ( t − [ z − 1 ] a ) τ ( l +1) ( t ) , l ≥ 1 . (189) 42 The dual p olynomials ¯ A ( l ) b , ¯ A ( l ) + b 0 ,b , and ¯ A ( l ) − a,b have the fol lowing τ -function r epr esentation ¯ A ( l ) b ( z ) = ¯ A (I ,a 2 ( b )) [ ~ ν 2 ( l ); ~ ν 1 ( l − 1)] ,b = z ν 2 ,b ( l ) − 1 ¯ τ ( l ) − b ( t + [ z − 1 ] b ) τ ( l +1) ( t ) , l ≥ 1 , (190) ¯ A ( l ) + b 0 ,b ( z ) = ¯ A (II ,b 0 ) [ ~ ν 2 ( l − 1)+ ~ e 2 ,b 0 ; ~ ν 1 ( l − 1)] ,b = z ν 2 ,b ( l − 1)+ δ b,b 0 − 1 ¯ τ ( l ) + b 0 , − b ( t + [ z − 1 ] b ) τ ( l ) ( t ) , l ≥ 1 , (191) ¯ A ( l ) − a,b ( z ) = ¯ A (I ,a ) [ ~ ν 2 ( l ); ~ ν 1 ( l ) − ~ e 1 ,a ] ,b = z ν 2 ,b ( l ) − 1 ¯ τ ( l ) − a, − b ( t + [ z − 1 ] b ) τ ( l +1) ( t ) , l ≥ 1 . (192) Pr o of. See App endix A Observ e that in the simple ladder deﬁned ab o ve ( ~ ν 1 , ~ ν 2 + ~ e 2 ,p 2 ) with l = | ~ ν 2 | = | ~ ν 1 | − 1 w e hav e ~ ν 1 ( l ) = ~ ν 1 , ~ ν 2 ( l − 1) = ~ ν 2 , ~ ν 1 ( l − 1) = ~ ν 1 − ~ e 1 ,p 1 , ~ ν 2 ( l ) = ~ ν 2 + ~ e 2 ,p 2 . F rom Prop osition 35 we get A (II ,p 1 ) [ ~ ν 1 ; ~ ν 2 ] ,a ( z ) = ε 1 , 1 ( p 1 , a ) z ν 1 ,a − 1 τ [ ~ ν 1 + ~ e 1 ,p 1 − ~ e 1 ,a a ; ~ ν 2 ] ( t − [ z − 1 ] a ) τ [ ~ ν 1 ; ~ ν 2 ] ( t ) , A (II ,a 0 ) [ ~ ν 1 − ~ e 1 ,p 1 + ~ e 1 ,a 0 ; ~ ν 2 ] ,a ( z ) = ε 1 , 1 ( a 0 , a ) z ν 1 ,a − δ a,p 1 + δ a,a 0 − 1 τ [ ~ ν 1 − ~ e 1 ,a + ~ e 1 ,a 0 ; ~ ν 2 ] ( t − [ z − 1 ] a ) τ [ ~ ν 1 ; ~ ν 2 ] ( t ) , A (I ,b ) [ ~ ν 1 ; ~ ν 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] ,a ( z ) = ε 2 , 1 ( b, a ) z ν 1 ,a − 1 τ [ ~ ν 1 + ~ e 1 ,p 1 − ~ e 1 ,a ; ~ ν 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] ( t − [ z − 1 ] a ) τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,p 2 ] ( t ) , (193) ¯ A (I ,p 2 ) [ ~ ν 2 + ~ e 2 ,p 2 ; ~ ν 1 − ~ e p 1 ] ,b ( z ) = ε 2 , 2 ( p 2 , b ) z ν 2 ,b + δ b,p 2 − 1 τ [ ~ ν 1 ; ~ ν 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] ( t + [ z − 1 ] b ) τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,p 2 ] ( t ) , ¯ A (II ,b 0 ) [ ~ ν 2 + ~ e 2 ,b 0 ; ~ ν 1 − ~ e 1 ,p 1 ] ,b = ε 2 , 2 ( b, b 0 ) z ν 2 ,b + δ b 0 ,b − 1 τ [ ~ ν 1 ; ~ ν 2 − ~ e 2 ,b + ~ e 2 ,b 0 ] ( t + [ z − 1 ] b ) τ [ ~ ν 1 ; ~ ν 2 ] ( t ) , ¯ A (I ,a ) [ ~ ν 2 + ~ e 2 ,p 2 ; ~ ν 1 − ~ e 1 ,a ] ,b = ε 2 , 1 ( b, a ) z ν 2 ,b + δ b,p 2 − 1 τ [ ~ ν 1 + ~ e 1 ,p 1 − ~ e 1 ,a ; ~ ν 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] ( t + [ z − 1 ] b ) τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,p 2 ] ( t ) . (194) W e no w present the τ -represen tation of the Cauc hy transforms of the linear forms. Prop osition 36. The Cauchy tr ansforms have the fol lowing τ -function r epr esentation ¯ C ( l ) a = z − ν 1 ,a ( l − 1) − 1 τ ( l +1) + a ( t + [ z − 1 ] a ) τ ( l +1) ( t ) , (195) C ( l ) b = z − ν 2 ,b ( l − 1) − 1 ¯ τ ( l +1) + b ( t − [ z − 1 ] b ) τ ( l ) ( t ) . (196) Pr o of. See App endix A W e ha ve the representation C (I ,p 2 ) [ ~ ν 2 + ~ e 2 ,p 2 ; ~ ν 1 − ~ e 1 ,p 1 ] ,a ( z ) = ε 1 , 1 ( a, p 1 ) z − ν 1 ,a − 1+ δ a,p 1 τ [ ~ ν 1 + ~ e 1 ,a ; ~ ν 2 + ~ e 2 ,p 2 ] ( t + [ z − 1 ] a ) τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,p 2 ] ( t ) , C (II ,p 1 ) [ ~ ν 1 ; ~ ν 2 ] ,b ( z ) = ε 2 , 2 ( b, p 2 ) z − ν 2 ,b − 1 τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,b ] ( t − [ z − 1 ] b ) τ [ ~ ν 1 ; ~ ν 2 ] ( t ) . Finally , w e consider the τ -function representation of the bilinear equation 43 Prop osition 37. The τ functions fulﬁl l the fol lowing biline ar r elation p 1 X a =1 I z = ∞ z ν 1 ,a ( k ) − ν 0 1 ,a ( l − 1) − 2 τ ( k ) ~ n 1 ,~ n 2 , − a ( t − [ z − 1 ] a ) τ ( l +1) ~ n 0 1 ,~ n 0 2 , + a ( t 0 + [ z − 1 ] a ) E a ( z )d z = p 2 X b =1 I z = ∞ z ν 0 2 ,b ( l ) − ν 2 ,b ( k − 1) − 2 ¯ τ ( k +1) ~ n 1 ,~ n 2 , + b ( t − [ z − 1 ] b ) ¯ τ ( l ) ~ n 0 1 ,~ n 0 2 , − b ( t 0 + [ z − 1 ] b ) ¯ E b ( z )d z . (197) Pr o of. Just consider (181) together with (187), (190), (195) and (196). This bilinear relation can also b e written as follows p 1 X a =1 ε 11 ( p 1 , a ) ε 0 11 ( p 1 , a ) I z = ∞ z ν 1 ,a − ν 0 1 ,a − δ a,p 1 − 2 τ [ ~ ν 1 + ~ e 1 ,p 1 − ~ e 1 ,a ; ~ ν 2 ] ( t − [ z − 1 ] a ) τ ( l +1) ~ ν 0 1 + ~ e 1 ,a ; ~ ν 0 2 + ~ e 2 ,p 2 ( t 0 + [ z − 1 ] a ) E a ( z )d z = p 2 X b =1 ε 22 ( p 2 , b ) ε 0 22 ( p 2 , b ) I z = ∞ z ν 0 2 ,b + δ b,p 2 − ν 2 ,b − 2 τ [ ~ ν 1 + ~ e 1 ,p 1 ; ~ ν 2 + ~ e 2 ,b ] ( t − [ z − 1 ] b ) τ ~ ν 0 1 ; ~ ν 0 2 + ~ e 2 ,p 2 − ~ e 2 ,b ] ( t 0 + [ z − 1 ] b ) ¯ E b ( z )d z . (198) That with the identiﬁcation m ∗ = ~ ν 1 + ~ e 1 ,p 1 , n ∗ = ~ ν 2 , m = ~ ν 0 1 and n = ~ ν 0 2 + ~ e 2 ,p 2 , up to signs, is the bilinear relation (41) in [6]. Ac kno wledgemen ts MM thanks economical supp ort from the Spanish Ministerio de Ciencia e Innov aci´ on, research pro ject FIS2008-00200 and UF thanks economical supp ort from the SFRH / BPD / 62947 / 2009, F unda¸ c˜ ao para a Ciˆ encia e a T ecnologia of P ortugal, PT2009-0031, Acciones Integradas Portugal, Ministerio de Ciencia e Innov aci´ on de Espa ˜ na, and MTM2009- 12740-C03-01. Ministerio de Ciencia e Innov aci´ on de Espa˜ na. MM rec kons diﬀerent and clarifying discussions with Dr. Mattia Caﬀasso, Prof. Pierre v an Moerb eke, Prof. Luis Mart ´ ınez Alonso and Prof. Da vid G´ omez-Ullate. The authors of this pap er are in debt with Prof. Guillermo L´ op ez Lagomasino who carefully read the manuscript and whose suggestions improv ed the pap er indeed. App endices A Pro ofs Pro of Prop osition 7 The orthogonalit y relations can b e recast into tw o alternative forms  S l, 0 S l, 1 · · · S l,l − 1       g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1      = −  g l, 0 g l, 1 · · · g l,l − 1  , l ≥ 1 , (199)  S l, 0 S l, 1 · · · S l,l − 1 S l,l       g 0 , 0 g 0 , 1 · · · g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l . . . . . . . . . g l, 0 g l, 1 · · · g l,l      =  0 0 · · · 0 ¯ S l,l  | {z } l + 1 comp onen ts , l ≥ 0 . (200) F rom (43) we get Q ( l ) = l X k =0 S l,k ξ ( k ) 1 = ξ ( l ) 1 −  g l, 0 g l, 1 · · · g l,l − 1  ( g [ l ] ) − 1 ξ [ l ] 1 use (199) (201) = ¯ S l,l  0 0 · · · 0 1  ( g [ l +1] ) − 1 ξ [ l +1] 1 . use (200) (202) 44 Cramer’s metho d solves (199) as follows S l,i = 1 det g [ l ] l − 1 X j =0 g l,j ( − 1) i + j +1 M ( l ) i,j = ( − 1) i + l M ( l +1) i,l det g [ l ] , (203) where M ( l ) i,j is the ( i, j )-minor of the truncated moment matrix g [ l ] deﬁned in (21). Therefore, Q ( l ) = 1 det g [ l ] l X i =0 ( − 1) i + l M ( l +1) i,l ξ ( i ) 1 = 1 det g [ l ] det           g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 ξ (0) 1 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 ξ (1) 1 . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 ξ ( l − 1) 1 g l, 0 g l, 1 · · · g l,l − 1 ξ ( l ) 1           , l ≥ 1 . The orthogonality relations for the dual system can b e written also in tw o alternative forms      g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1           ¯ S 0 0 ,l ¯ S 0 1 ,l . . . ¯ S l − 1 ,l      = − ( ¯ S l,l ) − 1      g 0 ,l g 1 ,l . . . g l − 1 ,l      , l ≥ 1 , (204)      g 0 , 0 g 0 , 1 . . . g 0 ,l g 1 , 0 g 1 , 1 . . . g 1 ,l . . . . . . . . . g l, 0 g l, 1 . . . g l,l           ¯ S 0 0 ,l ¯ S 0 1 ,l . . . ¯ S 0 l,l      =        0 0 . . . 0 1        , l ≥ 0 . (205) As b efore, (43) leads to the following expressions for the dual linear forms ¯ Q ( l ) = l X k =0 ¯ S 0 k,l ξ ( k ) 2 = ( ¯ S l,l ) − 1  ξ ( l ) 2 − ( ξ [ l ] 2 ) > ( g [ l ] ) − 1      g 0 ,l g 1 ,l . . . g l − 1 ,l       use (204) (206) = ( ξ [ l +1] 2 ) > ( g [ l +1] ) − 1        0 0 . . . 0 1        . use (205) (207) F rom (205) we obtain ¯ S 0 j,l =  g [ l +1] − 1  j,l = ( − 1) l + j M ( l +1) l,j det( g [ l +1] ) , j = 0 , . . . , l , (208) 45 and consequently ¯ Q ( l ) = l X j =0 ¯ S 0 j,l ξ ( j ) 2 = 1 det g [ l +1] l X j =0 ( − 1) l + j M ( l +1) l,j ξ ( j ) 2 = 1 det g [ l +1] det        g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 ,l . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 ,l ξ (0) 2 ξ (1) 2 · · · ξ ( l − 1) 2 ξ ( l ) 2        , l ≥ 0 . Pro of Prop osition 9 W e hav e C ( l ) b = 1 det g [ l ] l X k =0 ( − 1) k + l M ( l +1) k,l ∞ X k 2 = ν 2 ,b ( l − 1) z − k 2 − 1 Z x k 1 ( k ) w 1 ,a 1 ( k ) ( x ) w 2 ,b ( x ) x k 2 d µ ( x ) , whic h according to (20) recasts into C ( l ) b = 1 det g [ l ] l X k =0 ( − 1) k + l M ( l +1) k,l ¯ Γ ( l ) k,b , = 1 det g [ l ] det             g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 ¯ Γ ( l ) 0 ,b g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 ¯ Γ ( l ) 0 ,b . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 ¯ Γ ( l ) l − 1 ,b g l, 0 g l, 1 · · · g l,l − 1 ¯ Γ ( l ) l,b ,             , l ≥ 1 . (209) W e also obtain ¯ C ( l ) a = 1 det g [ l +1] l X k =0 ( − 1) l + k M ( l +1) l,k ∞ X k 1 = ν 1 ,a ( l − 1) z − k 1 − 1 Z x k 1 w 1 ,a ( x ) w 2 ,a 2 ( k ) x k 2 ( k ) d µ ( x ) , whic h can b e written as (20) ¯ C ( l ) a = 1 det g [ l +1] l X k =0 ( − 1) k + l M ( l +1) l,k Γ ( l ) k , = 1 det g [ l +1] det         g 0 , 0 g 0 , 1 · · · g 0 ,l − 1 g 0 ,l g 1 , 0 g 1 , 1 · · · g 1 ,l − 1 g 1 ,l . . . . . . . . . . . . g l − 1 , 0 g l − 1 , 1 · · · g l − 1 ,l − 1 g l − 1 ,l Γ ( l ) 0 ,a Γ ( l ) 1 ,a · · · Γ ( l ) l − 1 ,a Γ ( l ) l,a         , l ≥ 1 . (210) Pro of Prop osition 15 F rom (53) and (54) we deduce that ( ¯ C a ( z )) > A a 0 ( z 0 ) = ( χ ∗ 1 ,a ( z )) > χ 1 ,a 0 ( z 0 ) = δ a,a 0 z − z 0 , | z 0 | < | z | , ( C b ( z )) > ¯ A a 0 ( z 0 ) = ( χ ∗ 2 ,b ( z )) > χ 2 ,b 0 ( z 0 ) = δ b,b 0 z − z 0 , | z 0 | < | z | , ( ¯ C a ( z )) > C b ( z 0 ) = ( χ ∗ 1 ,a ( z )) > g χ ∗ 2 ,b ( z 0 ) . 46 The t wo ﬁrst relations imply the corresp onding equations in the Prop osition. F or the third we observe that from (10) w e get ( χ ∗ 1 ,a ( z )) > g χ ∗ 2 ,b 0 ( z 0 ) = Z ( χ ∗ 1 ,a ( z )) > ξ 1 ( x )( ξ 2 ( x )) > χ ∗ 2 ,b 0 ( z 0 )d µ ( x ) = Z ( χ ∗ 1 ,a ( z )) > χ 1 ,a ( x )( χ 2 ,b ( x )) > χ ∗ 2 ,b 0 ( z 0 ) w 1 ,a ( x ) w 2 ,b ( x )d µ ( x ) = Z 1 ( z − x )( z 0 − x ) w 1 ,a ( x ) w 2 ,b ( x )d µ ( x ) = − 1 z − z 0 Z  1 z − x − 1 z 0 − x  w 1 ,a ( x ) w 2 ,b ( x )d µ ( x ) . Pro of Prop osition 20 Using Deﬁnition 17 for the linear forms Q ( l ) + a and multiplying by ( ξ [ l ] 2 ( x )) > w e hav e Q ( l ) + a ( x )( ξ [ l ] 2 ( x )) > = ξ ( l + a ) 1 ( x )( ξ [ l ] 2 ( x )) > −  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  ( g [ l ] ) − 1 ξ [ l ] 1 ( x )( ξ [ l ] 2 ( x )) > , in tegrating b oth sides we get Z Q ( l ) + a ( x )( ξ [ l ] 2 ( x )) > d µ ( x ) = Z ξ ( l + a ) 1 ( x )( ξ [ l ] 2 ( x )) > d µ ( x ) −  g l + a , 0 · · · g l + a ,l − 1  ( g [ l ] ) − 1 Z ξ [ l ] 1 ( x )( ξ [ l ] 2 ( x )) > d µ ( x ) = Z ξ ( l + a ) 1 ( x )( ξ [ l ] 2 ( x )) > d µ ( x ) −  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  ( g [ l ] ) − 1 g [ l ] =  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  −  g l + a , 0 g l + a , 1 · · · g l + a ,l − 1  = 0 , that written comp onen twise gives the following orthogonality relations Z R Q ( l ) + a ( x ) w 2 ,a 2 ( k ) ( x ) x k 2 ( k ) d µ ( x ) , k = 0 , . . . , l − 1 , or equiv alently Z R Q ( l ) + a ( x ) w 2 ,b ( x ) x k d µ ( x ) = 0 , 0 ≤ k ≤ ν 2 ,b ( l − 1) − 1 , b = 1 , . . . , p 2 . Notice that, A ( l ) + a,a is monic and deg A ( l ) + a,a ( x ) = k 1 ( l + a ) but A ( l ) + a,a 0 with a 6 = a 0 satisfy deg A ( l ) + a,a 0 ≤ k 1 (( l − 1) − a 0 ). This means that the set of p olynomials A ( l ) + a,a 0 ( x ) hav e degrees determined by ~ ν 1 ( l − 1) + ~ e 1 ,a and a normalization with resp ect to the a -th comp onen t of type I I; i.e, Q ( l ) + a = Q (II ,a ) [ ~ ν 1 ( l − 1)+ ~ e 1 ,a ; ~ ν 2 ( l − 1)] . In a similar wa y , the asso ciated linear forms Q ( l ) − b ( x ) solve a mixed m ultiple orthogonal problem that can b e obtained as follows. F rom Deﬁnition 17 and multiplying by ( ξ [ l ] 2 ( x )) > w e get Q ( l ) − b ( x )( ξ [ l +1] 2 ( x )) > = e > ¯ l − b ( g [ l +1] ) − 1 ξ [ l +1] 1 ( x )( ξ [ l +1] 2 ( x )) > , in tegrating b oth sides Z R Q ( l ) − b ( x )( ξ [ l +1] 2 ( x )) > d µ ( x ) = e > ¯ l − b ( g [ l +1] ) − 1 Z R ξ [ l +1] 1 ( x )( ξ [ l +1] 2 ( x )) > d µ ( x ) = e > ¯ l − b , and written comp onen twise Z R Q ( l ) − b ( x ) x k 2 ( k ) w 2 ,a 2 ( k ) ( x )d µ ( x ) = δ k, ¯ l − b , k = 0 , · · · , l, that is equiv alent to Z R Q ( l ) − b ( x ) w 2 ,b ( x ) x k d µ ( x ) = δ k, ¯ l − b , 0 ≤ k ≤ ν 2 ,b ( l ) − 1 , b = 1 , . . . , p 2 . 47 Hence, the set A ( l ) − b,a 0 is a type I normalized to the b -th component solution for a mixed m ultiple orthogonalit y problem; the degrees satisfy deg A ( l ) − b,a 0 ≤ l − a 0 . Moreov er, the fact that the last orthogonalit y condition in the b -th comp onen t is missing gives the identiﬁcation Q ( l ) − b = Q (I ,b ) [ ~ ν 1 ( l ); ~ ν 2 ( l ) − ~ e 2 ,b ] . Using Deﬁnition 17 and multiplying by ξ [ l ] 1 ( x ) we hav e ¯ Q ( l ) + b ( x ) ξ [ l ] 1 ( x ) =  ξ [ l ] 1 ( x ) ξ ( ¯ l + b ) 2 ( x ) − ξ [ l ] 1 ( x )( ξ [ l ] 2 ( x )) > ( g [ l ] ) − 1      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b       , and integrating b oth sides Z R ¯ Q ( l ) + b ( x ) ξ 1 ( x ) [ l ] d µ ( x ) =  Z ξ [ l ] 1 ( x ) ξ ( ¯ l + b ) 2 ( x )d µ ( x ) − Z ξ [ l ] 1 ( x )( ξ [ l ] 2 ( x )) > d µ ( x )( g [ l ] ) − 1      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b       =  Z ξ [ l ] 1 ( x ) ξ ( ¯ l + b ) 2 ( x )d µ ( x ) − ( g [ l ] )( g [ l ] ) − 1      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b       =      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b      −      g 0 , ¯ l + b g 1 , ¯ l + b . . . g l − 1 , ¯ l + b      = 0 , that comp onen twise leads to the following orthogonality relations Z R ¯ Q ( l ) + b ( x ) w 1 ,a 1 ( k ) ( x ) x k 1 ( k ) d µ ( x ) = 0 , k = 0 , . . . , l − 1 , or alternatively Z ¯ Q ( l ) + b ( x ) w 1 ,a ( x ) x k d µ ( x ) = 0 , 0 ≤ k ≤ ν 1 ,a ( l − 1) − 1 , a = 1 , . . . , p 1 . Notice that, ¯ A ( l ) + b,b is monic and deg ¯ A ( l ) + b,b = k 2 ( ¯ l + b ) but ¯ A ( l ) + b,b 0 with b 6 = b 0 satisfy deg A ( l ) + b,b 0 ≤ k 1 (( l − 1) − b 0 ). This means that the p olynomials ¯ A ( l ) + b,b 0 ha ve degrees determined b y ~ ν 2 ( l − 1) + ~ e 2 ,b and a normalization with respect to the b -th comp onen t of type I I; i.e, ¯ Q ( l ) + b = ¯ Q (II ,b ) [ ~ ν 2 ( l − 1)+ ~ e 2 ,b ; ~ ν 1 ( l − 1)] . Finally , w e obtain the orthogonality relations for the linear forms ¯ Q ( l ) − a ( x ). F rom the deﬁnition w e get ξ [ l +1] 1 ( x ) ¯ Q ( l ) − a ( x ) = ξ [ l +1] 1 ( x )( ξ [ l +1] 2 ( x )) > ( g [ l +1] ) − 1 e l − a , and integrating b oth sides Z R ξ [ l +1] 1 ( x ) ¯ Q ( l ) − a ( x )d µ ( x ) =  Z R ξ [ l +1] 1 ( x )( ξ [ l +1] 2 ( x )) > d µ ( x )  ( g [ l +1] ) − 1 e l − a = e l − a , and comp onen twise that means Z R ¯ Q ( l ) − a ( x ) x k 1 ( k ) w 1 ,a 1 ( k ) ( x )d µ ( x ) = δ k,l − a , k = 0 , · · · , l, or equiv alently Z R ¯ Q ( l ) − a ( x ) x k w 1 ,a ( x )d µ ( x ) = δ k,l − a , 0 ≤ k ≤ ν 1 ,a ( l ) − 1 , a = 1 , . . . , p 1 , so the set ¯ A ( l ) − a,b 0 is a t yp e I normalized to the a -th comp onent solution for a mixed m ultiple orthogonality problem. The degrees satisfy deg A ( l ) − a,b 0 ≤ l − b 0 ; and therefore we conclude that ¯ Q ( l ) − b = Q (I ,b ) [ ~ ν 2 ( l ); ~ ν 1 ( l ) − ~ e 1 ,a ] . 48 Pro of of Prop osition 32 T aking + and − parts in (170) we obtain p 1 X a =1 L j a = p 1 X a =1 ( L j a ) + + p 1 X a =1 ( L j a ) − = p 1 X a =1 ( L j a ) + + p 2 X b =1 ( ¯ L j b ) − = p 1 X a =1 B j,a + p 2 X b =1 ¯ B j,b = p 2 X b =1 ¯ L j b . Using Lax equations and observing that L a L a 0 = L a 0 L a and ¯ L b ¯ L b 0 = ¯ L b 0 ¯ L b w e hav e the following symmetries for the Lax op erators  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  L a 0 = h p 1 X a =1 B j,a + p 2 X b =1 ¯ B j,b , L a 0 i = p 1 X a =1 [ L j a , L a 0 ] = 0 ,  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  ¯ L b 0 = h p 1 X a =1 B j,a + p 2 X b =1 ¯ B j,b , ¯ L b 0 i = p 2 X b =1 [ ¯ L j b , ¯ L b 0 ] = 0 . F rom (127) we conclude that the m ultiple orthogonal p olynomials and their duals are also inv ariant  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  A a 0 =  p 1 X a =1 B j,a + p 2 X b =1 ¯ B j,b − x j  A a 0 =  J j − x j  A a 0 = 0 ,  p 1 X a =1 ∂ ∂ t j,a + p 2 X b =1 ∂ ∂ ¯ t j,b  ¯ A b 0 = −  p 1 X a =1 B j,a + p 2 X b =1 ¯ B j,b − x j  > ¯ A b 0 = −  J j − x j  > ¯ A a 0 = 0 . Pro of of Prop osition 33 W e just follow the follo wing c hain of identities W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 1 ,~ n 1 ( W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 = W ~ n 1 ,~ n 2 ( t, s ) π > ~ n 0 1 ,~ n 1 g ~ n 0 1 ,~ n 0 2 ( ¯ W ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 = S ~ n 1 ,~ n 2 ( t, s ) W 0 ,~ n 1 ( t, s ) π > ~ n 0 1 ,~ n 1  Z ξ ~ n 0 1 ( x ) ξ > ~ n 0 2 ( x )d µ ( x )  ( W 0 ,~ n 0 2 ( t 0 , s 0 )) − 1 ( ¯ S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 = S ~ n 1 ,~ n 2 ( t, s ) W 0 ,~ n 1 ( t, s )  Z ξ ~ n 1 ( x ) ξ > ~ n 0 2 ( x )d µ ( x )  ( W 0 ,~ n 0 2 ( t 0 , s 0 )) − 1 ( ¯ S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 = S ~ n 1 ,~ n 2 ( t, s )  Z ξ ~ n 1 ( x, t, s ) ξ > ~ n 0 2 ( x, t 0 , s 0 )d µ ( x )  ( ¯ S ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 = Z ( S ~ n 1 ,~ n 2 ( t, s )( ξ ~ n 1 ( x, t, s ))( ¯ S > ~ n 0 1 ,~ n 0 2 ( t 0 , s 0 )) − 1 ξ ~ n 0 2 ( x, t 0 , s 0 )) > d µ ( x ) = Z R Q ~ n 1 ,~ n 2 ( x, t, s ) ¯ Q > ~ n 0 1 ,~ n 0 2 ( x, t 0 , s 0 )d µ ( x ) , where ξ ~ n 1 ( x, t, s ) and ξ ~ n 0 2 ( x, t 0 , s 0 ) represent the vectors of w eighted monomials but with evolv ed w eights. Pro of of Proposition 35 T o ﬁnd the τ -function of the m ultiple orthogonal representation we ﬁrst need t wo lemmas Lemma 10. L et R ( j ) b e the j -th r ow of τ ( l ) ( t ) and R ( j ) z the j -th r ow of τ ( l ) ( t − [ z − 1 ] a ) , then R ( j ) z = R ( j ) − δ a 1 ( j ) ,a z − 1 R ( j 0 ) , (211) wher e j 0 = j + 1 if r 1 ( j ) < n 1 ,a − 1 , but j 0 = j + ( | ~ n 1 | − n 1 ,a ) + 1 if r 1 ( j ) = n 1 ,a − 1 . This is also valid for τ ( l ) − a , τ ( l ) + a, − a 0 and for τ ( l ) − b, − a 0 . L et now b e C ( j ) the j-th c olumn of ¯ τ ( l ) and C ( j ) z the j -th c olumn of ¯ τ ( l ) ( t + [ z − 1 ] b ) , then C ( j ) z = C ( j ) − δ a 2 ( j ) ,b z − 1 C ( j 0 ) , (212) wher e j 0 = j + 1 if r 2 ( l ) < n 2 ,b − 1 but j 0 = j + ( | ~ n 2 | − n 2 ,b ) + 1 if r 2 ( j ) = n 2 ,b − 1 . This is also valid for ¯ τ ( l ) − b , ¯ τ ( l ) + b, − b 0 and for ¯ τ ( l ) − a, − b 0 . Pr o of. It follows directly from (141) and (142). 49 Let us recall the skew m ulti-linear character of determinan ts and the consequent form ulation in terms of w edge pro ducts of cov ectors. Observ e that Lemma 11. Given a set of c ove ctors { r 1 , . . . , r n } it c an b e shown that n ^ j =1 ( z r j − r j +1 ) = n +1 X j =1 ( − 1) n +1 − j z j − 1 r 1 ∧ r 2 ∧ · · · ∧ ˆ r j ∧ · · · ∧ r n +1 , (213) wher e the notation ˆ r j me ans that we have er ase d the c ove ctor r j in the we dge pr o duct r 1 ∧ · · · ∧ r n +1 . Pr o of. It can b e done directly by induction. The pro of of Proposition 35 relies on Lemma 10, Lemma 11, Corollary 1 and Prop osition 22. First let’s fo cus on (187); it is clear that z ν 1 ,a ( l ) − 1 τ ( l ) − a ( t − [ z − 1 ] a ) expands in z according to (211) for τ ( l ) − a and to (213). No w n = k 1 ( l − a ) and the cov ectors r j should b e taken equal to those rows R ( j ) with a 1 ( j ) = a . Observ e that there are only k 1 ( l − a )(= ν 1 ,a ( l ) − 1) rows which are non-trivially transformed. In this form w e get the identiﬁcation of (49) with (187), where the terms corresponding to the w edge with one co v ector deleted corresp onds to the minors M [ l +1] j,l . Now, lo oking to (188) and (189) we expand again in z and use the same technique based on (211) for τ ( l ) + a, − a 0 and τ ( l ) − b, − a and (213). These allow to link (104) to (188) and (106) to (189). T o pro ve (190) we pro ceed similarly . Lo oking at (212) for ¯ τ ( l ) − b observ e that there are only k 2 ( ¯ l − b )(= ν 2 ,b ( l ) − 1) columns which are non-trivially transformed. No w, recalling (50) and using (213) but with r j b eing the columns C ( j ) , suc h that a 2 ( j ) = b , and n = k 2 ( ¯ l − b ), w e get the desired result. Finally for (191) and (192) w e expand around z to see the equiv alence b et ween (107) and (191) and the equiv alence b etw een (105) and (192). Pro of of Prop osition 36 W e need the following t wo lemmas: Lemma 12. L et R ( j ) b e the j -th r ow of g [ l +1] + a and R ( j ) z the j -th r ow of g [ l +1] + a ( t + [ z − 1 ] a ) , we get R ( j ) z = R ( j ) + δ a 1 ( j ) ,a ∞ X j 0 =1 z − k 1 ( j 0 ) R ( j + j 0 ) δ a 1 ( j + j 0 ) ,a . (214) L et C ( j ) b e the j -th c olumn of ¯ g [ l +1] + b and C ( j ) z the j -th c olumn of ¯ g [ l +1] + b ( t − [ z − 1 ] b ) , then (142) gives C ( j ) z = C ( j ) + δ a 2 ( j ) ,b ∞ X j 0 =1 z − k 2 ( j 0 ) C ( j + j 0 ) δ a 2 ( j + j 0 ) ,b . (215) Pr o of. F or the ﬁrst equalit y insert the expansion  1 − x z  − 1 = ∞ X k =0 x k z k in to (142). The other equation is prov en similarly . Lemma 13. The fol lowing identity n ^ j =1  ∞ X i =0 r j + i z − i  = r 1 ∧ · · · ∧ r n − 1 ∧  ∞ X i =0 r n + i z − i  (216) holds. Pr o of. Use induction in n . Finally Prop osition 36 is prov en using (210) and (209). 50 B Discrete ﬂo ws asso ciated with binary Darb oux transformations When the supp orts of the measures are not b ounded from b elo w, (in which case the new “w eights” (138) do not hav e in general a deﬁnite sign and therefore should not b e considered as such), there is an alternative form of constructing discrete ﬂows which preserve the p ositiveness/negativ eness of the measures. The construction is based in the previous one, but no w the shift is the comp osition of tw o consecutive shifts asso ciated with the pair λ a ( n ) and λ a ( n + 1), b eing complex num b ers conjugate to each other; i.e., w e consider Deﬁnition 27. We deﬁne a deforme d moment matrix g ( s ) = D 0 0 ( s ) g ( ¯ D 0 0 ( s )) − 1 . (217) with D 0 0 := p 1 X a =1 D 0 0 ,a , D 0 0 ,a :=      Q s a n =1  | λ a ( n ) | 2 Π 1 ,a − 2 Re( λ a ( n ))Λ 1 ,a + Λ 2 1 ,a  , s a > 0 , Π 1 ,a , s a = 0 , Q | s a | n =1  | λ a ( − n ) | 2 Π 1 ,a − 2 Re( λ a ( − n ))Λ 1 ,a + Λ 2 1 ,a  − 1 , s a < 0 , (218) ( ¯ D 0 0 ) − 1 := p 2 X b =1  ( ¯ D 0 ( s ) 0 ) − 1  b ,  ( ¯ D 0 ( s ) 0 ) − 1  b :=      Q ¯ s b n =1 ( | ¯ λ b ( n ) | 2 Π 2 ,b − 2 Re( ¯ λ b ( n ))Λ > 2 ,b ) + (Λ > 2 ,b ) 2 )  ¯ s b > 0 , Π 2 ,b , ¯ s b = 0 , ( Q ¯ s b n =1  | ¯ λ b ( − n ) | 2 Π 2 ,b − 2 Re( ¯ λ b ( − n ))Λ > 2 ,b ) + (Λ > 2 ,b ) 2  − 1 , ¯ s b < 0 . (219) Prop osition 38. The pr eviously deﬁne d deforme d moment matrix c orr esp onds to a moment matrix with the fol lowing p ositive/ne gative evolve d weights w 1 ,a ( s, x ) = D 0 a ( x, s a ) w 1 ,a ( x ) , D 0 a :=      Q s a n =1 | x − λ a ( n ) | 2 , s a > 0 , 1 , s a = 0 , Q | s a | n =1 | x − λ a ( − n ) | − 2 , s a < 0 , w 2 ,b ( s, x ) = ¯ D 0 b ( x, ¯ s b ) − 1 w 2 ,b ( x ) , ( ¯ D 0 b ) − 1 :=      Q ¯ s b n =1 | x − ¯ λ b ( n ) | 2 , ¯ s b > 0 , 1 , ¯ s b = 0 , Q | ¯ s b | n =1 | x − ¯ λ b ( − n ) | − 2 , ¯ s b < 0 . (220) Pro ceeding as in the previous case Deﬁnition 28. We intr o duc e q 0 a := I − Π 1 ,a ( I − | λ a ( s a + 1) | 2 ) − 2 Re( λ a ( s a + 1))Λ 1 ,a + Λ 2 1 ,a , ¯ q 0 b := I − Π 2 ,b ( I − | ¯ λ b ( ¯ s b + 1) | 2 ) − 2 Re( ¯ λ b ( ¯ s b + 1))(Λ > 2 ,b ) + (Λ > 2 ,b ) 2 , (221) and δ 0 a := I − C a ( I − | λ a ( s a + 1) | 2 ) − 2 Re( λ a ( s a + 1)) L a + L 2 a , ¯ δ 0 b := I − ¯ C b ( I − | ¯ λ b ( ¯ s b + 1) | 2 ) − 2 Re( ¯ λ b ( ¯ s b + 1)) ¯ L b + ¯ L 2 b . (222) The wa ve and adjoint w av e functions now hav e the form Ψ ( k ) a ( z , s ) = A ( k ) a ( z , s ) D 0 a ( z , s a ) , ( ¯ Ψ ∗ b ) ( k ) ( z , s ) = ¯ A ( k ) b ( z , s ) ¯ D 0 b ( z , ¯ s b ) − 1 , (223) and the expressions (161)-(162) still hold. 51 If we introduce ω 0 a and ¯ ω 0 b as in (154) but replacing δ by δ 0 , the equations (155)-(158) hold true by replacemen t of ω by ω 0 . Now, the form ω 0 diﬀers from (164) as now we hav e ω 0 a = ω 0 a, 0 Λ 2( | ~ n 1 |− n 1 ,a +1) + · · · + ω 0 a, 2( | ~ n 1 |− n 1 ,a +1) , ¯ ω 0 b = ¯ ω 0 b, 0 (Λ > ) 2( | ~ n 2 |− n 2 ,b +1) + · · · + ¯ ω 0 b, 2( | ~ n 2 |− n 2 ,b +1) . (224) With the deﬁnition of γ 0 a,a 0 ( s, x ) := (1 − δ a,a 0 (1 − | x − λ a ( s a + 1) | 2 ) , γ 0 b,b 0 ( s, x ) := (1 − δ b,b 0 (1 − | x − ¯ λ b ( ¯ s b + 1) | 2 ) , (225) w e hav e that Prop osition 39. 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Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy

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