Connection preserving deformations and $q$-semi-classical orthogonal polynomials

Connection preserving deformations and q -semi-classica l orthogonal p olynomials Christopher M. Ormero d, N. S. Wi tte and P ete r J. F orrester Abstract. W e presen t a framework for the study of q -diﬀerence equations satisﬁed by q -semi-classical orthogonal systems. As an example, we ident ify the q -diﬀerence equation satisﬁed by a deformed version of the l ittle q -Jacobi polynomials as a gauge transformation of a sp ecial case of the asso ciated linear problem f or q -P VI . W e obtain a parameterization of the associated linear problem in terms of orthog onal p olynomial v ar iables and ﬁnd t he relation betw een this parameterization and that of Jim bo and Sak ai. 1. Intr o duction Mono dromy r epresentations hav e been a central element in the study of integrable sys tems [ 5 ]. A pioneering step was the para meterization o f the condition that a linear second order diﬀeren tial equation with four regular singularities { 0 , t, 1 , ∞} has mono dromy indep endent of t , in terms of the sixth Painlev ´ e equa tio n P VI [ 21, 22 ]. This theory was elabo rated upo n by Garnier [ 2 5 ] and Schlesinger [ 59 ], and culminated in the 1980’s with the studies of the Kyoto School [ 37, 38, 3 9 ]. A contempora r y p ersp ective of the theory can b e found in the monogr aphs [ 3 5 ] and [ 18 ]. F or a matrix linear diﬀerential e q uation of the form (1.1) d d x Y ( x ) = A ( x ) Y ( x ) , where A ( x ) = X i A i x − α i , one expec ts the general solution to b e m ultiv a lued w ith branch p oints lo cated at α = { α i } . By ev aluating a solution on any ele men t o f the homotopy clas s es of closed lo ops, [ γ ], in some manner around a selection of the p oles, one obtains the equatio n Y ( γ (1)) = Y ( γ (0)) M [ γ ] . This relates so lutions on diﬀerent sheets of a Riemann surface. The set { M [ γ ] : γ : [0 , 1 ] → C } is a repr esen tation of the fundamental gro up of the c omplimen t of the p oles, Γ = π 1 ( CP 1 \ { α } ). The a im is to deform the linear system with r espec t to a chosen deformation parameter , t , so that the r e presentation of Γ doe s not dep end on t . In the theory of mono dromy preserving deforma tio ns, a na tural choice of para meters are the p oles of A . This leads to the 1 2 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER classical Schlesinger’s equations [ 59 ] ∂ A i ∂ α j = [ A i , A j ] α i − α j , i 6 = j, ∂ A i ∂ α i = − X j 6 = i [ A i , A j ] α i − α j . Given a 2 × 2 linea r system with four p oles, { 0 , t, 1 , ∞} , and zero of A 12 ( x ) at y , imp osing the isomono dromic prop erty in the v aria ble t requir es tha t y satisﬁes P VI [ 21 ]. The relationship of the theory of monodro m y preserving deformations to orthogonal polynomials ar ises b ecause under cer tain conditions on the o rthogonality meas ur e the po lynomials and their ass o cia ted functions form an isomono dromic sy stem, albeit one with a particular restriction. Under fairly g eneral co nditions the deriv ative of each po lynomial in the system is expressible in terms of linea r combinations of other mem ber s of the orthogo nal po lynomial system, a n obser v ation ﬁrs t made by L aguerre [ 45 ]. The conditions fo r when this is the cas e hav e b een given in some generality by Bonan a nd Clark[ 12 ], and by Bauldry [ 7 ]. In particular , a semi-classical weigh t w ill satisfy these conditions and the three term recurr ence tells us tha t we ma y express the deriv ative of the p olynomial in the system of orthogo nal p olynomia ls as a ra tio nal linear combination of the p olynomial itself and the previo us po lynomial in the system. The rationality of this linear problem means that such o rthogonal po ly nomial systems satisfy a linea r problem of the form (1.1). The notion of a s emi-classical w eig ht o r linea r functiona l was intro duced b y Mar oni [ 50 ] as an attempt to characterize the cla ssical ortho gonal p olynomials in a coherent framework and g uide the quest of loo k ing for sy stems beyond this class. By appro priately extending the work of La g uerre, Magnus [ 49 ] was able to show that a semi- classical, deformed orthog onal p olynomial system (1.1) para meterized a sp ecial c ase o f the mono dro m y preserving deformation co ns idered by F uchs [ 21 ]. This a llo ws o ne to ex pr ess s pecial solutions o f P VI in terms of co eﬃcients of orthogo nal polynomial systems. Conv er sely , this also allows key quantities rela ting to orthogo nal polynomia ls to be expressed in terms of solutio ns of P VI . In addition to the v a r ious determinantal s olutions of integrable systems provided by the theory o f o rthogonal p olynomia ls , the applicatio n o f integrable systems to orthogo nal p olynomials hav e re s ulted in adv ance s in the calculation of v a rious sta tistics of int erest in ra ndom matrix theor y (see e.g . [ 19 ]). F or q -diﬀere nce equations, an analogue o f the theor y of monodr om y pr eserving defo rmations is the theory of connection preser ving deformatio ns [ 40 ]. The linear problem of interest is given by the n × n matrix equation (1.2) Y ( qx ) = A ( x ) Y ( x ) , where A ( x ) = A 0 + A 1 x + . . . + A m x m . Instead of the bas ic information b eing contained in the rela tion betw een the v a lue of solutions o n diﬀeren t sheets o f the Riemann surface, the v aria bles of interest are asso ciated with the rela tio n betw een t wo fundamen tal solutions, Y 0 and Y ∞ , which a re holomo rphic functions at 0 and ∞ r espe c tively . Much of the theory concer ning the existence o f these s olutions ha s remained r elatively unc hanged since the pioneering days o f Birkhoﬀ and his followers [ 1, 10, 14 ]. If these s olutions exis t, then one may mer omorphically co n tin ue these solutions on C , and furthermor e form the CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 3 connection matrix, P ( x ), sp eciﬁed by Y 0 ( x ) = Y ∞ ( x ) P ( x ) , which is quasi-p erio dic in x [ 10 ]. F rom the Galois theor y of q -diﬀerence equations, whic h primarily considers a classiﬁcation of problems of the for m (1.2) , we know that the ent ries of P ( x ) are expressible in ter ms o f elliptic theta functions [ 58 ]. In the same manner as mono dromy preserving defor ma tions, one may c o nsider a deformation of (1.2) that preserves the connection matrix . An a ppr opriate choice of deformation par a meter tur ns out to b e the ro ots o f the determinant of A and the eigenv a lues of A 0 and A m . B y consider ing a 2 × 2 linear sys tem with m = 2 and cho osing the de fo rmation par ameter to be prop ortiona l to t wo of the ro ots of the deter minan t and the tw o eigenv alues of A 0 , Jimbo and Sak ai [ 40 ] show ed that the connection preserving defo rmation was equiv alent to a seco nd order q -diﬀerence equation admitting the sixth Painlev´ e equatio n as a co n tin uum limit. W e ha ve r emarked that semi-cla ssical ortho gonal p olynomial systems giv e rise to monodr om y pres erving de- formations rela ting to Painlev´ e equa tio ns. A natura l problem then is to in vestigate the r elationship b et w een q - semi-classica l orthogonal p olynomia l s ystems, connection preser ving defor mations, and the q -Painlev´ e equations . A nu mber of diﬀerent approa c hes to constructing isomo no dro mic a nalogues for the diﬀere nce, q -diﬀerence and elliptic equations of the Sak ai Scheme [ 57 ] have b een pro pos e d recen tly [ 3, 4, 55, 6 5 ], whic h diﬀer in v ary ing degrees from what we o ﬀer her e. One other work which is close to the spir it o f the present work is that of Bia ne [ 9 ]. How ever there is a history of studies into q -semi-cla ssical orthogo nal p olynomial systems which was not motiv a ted by the ab ov e consider ations. Shortly after the in tro duction of the semi-classica l concepts Magnus extended this to the q -diﬀerence s ystems and in fact to the most general type of divided diﬀerence ope r ators on non-unifor m lattices in a pionee ring s tudy [ 48 ]. In addition Ma roni and his co -work e rs have made ex tensions to diﬀerence and q -diﬀerence systems in a series of w orks [ 43, 51, 42, 52, 27 ]. These later authors hav e successfully repro duced parts o f the classic a l Askey T ablea ux (which was achieved most fully by Magnus at the level of the Askey-Wilson po lynomials) howev er the application o f their theor etical to ols b eyond the classical cases have in v a riably b een made to sp ecialised or degener ate cases a nd failed to ma k e contact with the dis crete and q -Painlev ´ e equations. A slightly diﬀerent metho dology has been the a pproach of Ismail and colla b ora tors [ 3 3, 31, 16, 32 ], who have derived diﬀerence and q -diﬀerence equatio ns for orthog onal po lynomials with res pect to weigh ts mor e genera l than the s emi-classical class, m uch in the spirit o f the Bonan and Clark and Bauldry studies, and so the matrix A ( x ) is no longer rational. This approach has not been applied to systems be yond the classical Askey T ablea ux, and consequently not made co n tact with the discr ete Painlev´ e systems. The most r e c en t work of Biane [ 9 ], and o f V an Assche a nd co-workers [ 64, 11 ] has addressed some o f the shortco mings disc us sed ab ov e, howev er while these authors hav e uncovered the sp ectra l structures o f the theor y they have yet to elucidate the defor ma tion str uctures required. It is our inten tion to complete this task by laying o ut the defor mation structures . Our contributions in this pap er ar e to ﬁrst for m ulate a n extension o f the classic al work of La guerre for ﬁnding diﬀerential equations satisﬁed by or tho gonal p olynomia ls , when the diﬀerential op erator is the q - diﬀerence (1.3) D q,x i f ( x 1 , . . . , x n ) = f ( x 1 , . . . , x n ) − f ( x 1 , . . . , q x i , . . . x n ) x i (1 − q ) . 4 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER This is done in Section 4, after ﬁrst having introduced preliminary material from or tho gonal p olynomial theory , and connection preserving deformations in Sections 2 and 3 resp ectively . In Sectio n 5 we apply this extension to the sp eciﬁc case of a defor mation of the little q -Jac obi p olynomials [ 2 ]. W e give a parameteriza tion of the asso ciated linear pr oblem in ter ms of v a riables relating to the orthogona l p olynomia l system. Ho wev er, this contains redundant v ariables, and in fact a set of thre e na tural co ordinates can b e identiﬁed which suﬃce to pa rameterize the linear problem. When wr itten in terms o f the natur al co ordinates, the linear problem implies the q -P VI equations y 1 ˆ y 1 = a 7 a 8 ( y 2 − a 1 t )( y 2 − a 2 t ) ( y 2 − a 3 )( y 2 − a 4 ) , (1.4a) y 2 ˆ y 2 = a 3 a 4 ( ˆ y 1 − a 5 t )( ˆ y 1 − a 6 t ) ( ˆ y 1 − a 7 )( ˆ y 1 − a 8 ) , (1.4b) where y i = y i ( t ) and ˆ y i = y i ( q t ). W e show that this has the co nsequence of implying the τ –functions hav e determi- nantal so lutions in terms of Hankel deter minan ts of the moments of the little q - Jacobi weigh t. Also we show that the three term recurr e nc e , written in terms of the natural co ordinates, manifests itself as a B¨ acklund tra nsformation which relate to a translationa l co mponent o f the extended aﬃne W eyl group of type D (1) 5 . Thro ug hout we shall assume that q is a ﬁxed co mplex num b er such that 0 < | q | < 1. 2. Orthogo n al po lynomials Our s tarting p oint is a s equence of moments, { µ k } ∞ k =0 . F rom this we deﬁne a linear functional, L , on the space of p olynomials, where L ( x k ) = µ k . An orthog o nal p olynomial system is a s equence of p olynomials, { p n } ∞ n =0 , such that p m is a p olynomial o f exa ct degree m and L ( p i p j ) = δ ij . (2.1) In other words, these p olynomials are orthonorma l with resp ect to the giv en linear functional. This condition deﬁnes the co eﬃcients o f p n for a ll n s o long as the Hank el determina nts cons is ting of the moment s µ 0 , . . . , µ 2 n , given in (2.3), do not v a nish [ 17, 61 ]. In the case of the c la ssical co ntin uous orthogo nal poly nomials, this linea r functional, L , is typically some integral of the multiplication of the argument with some weigh t function ov er some supp ort. Linear functionals asso ciated with discrete orthogona l p olynomials are spe ciﬁed b y a weigh ted sum, suc h as Jackson’s q -integral [ 28, 36, 62, 6 3 ]. An y ortho gonal p olynomia l s y stem where (2.1) holds satisﬁes the clas sical three term re c urrence r e lation, g iv en by (2.2) a n +1 p n +1 = ( x − b n ) p n − a n p n − 1 . W e parameterize the co eﬃcients of these po lynomials by p n ( x ) = γ n x n + γ n, 1 x n − 1 + γ n, 2 x n − 2 + . . . + γ n,n . CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 5 It is p ossible to determine a ll the co eﬃcients, and hence the a n and b n , in terms of the µ k ’s [ 17, 61 ]. W e s et (2.3) ∆ n = det            µ 0 µ 1 . . . µ n − 1 µ 1 µ 2 . . . µ n . . . . . . . . . µ n − 2 µ n − 1 . . . µ n − 1 µ n . . . µ 2 n − 2            , for n ≥ 1 , with ∆ 0 = 1, and (2.4) Σ n = det         µ 0 µ 1 . . . µ n − 2 µ n µ 1 µ 2 . . . µ n − 1 µ n +1 . . . . . . . . . . . . . . . µ n − 1 µ n . . . µ 2 n − 3 µ 2 n − 1         , for n ≥ 1 , where Σ 0 = 0. Then we hav e γ 2 n = ∆ n ∆ n +1 , (2.5a) a 2 n = ∆ n − 1 ∆ n +1 ∆ 2 n , (2.5b) b n = Σ n +1 ∆ n +1 − Σ n ∆ n , (2.5c) as g iven in [ 17, 6 1 ]. Given a sequence of v a lid moments, one may deﬁne the Stieltjes function f = ∞ X n =0 µ n x − n − 1 . W e deﬁne the as s ocia ted p olynomials and asso ciated functions by the formula f p n = φ n − 1 + ǫ n , where φ n − 1 is a po lynomial and ǫ n is the r emainder. The ortho gonality condition implies that ǫ n ∼ γ − 1 n x − n − 1 + O ( x − n − 2 ). In fact, by using (2.2), it is p ossible to ﬁnd the lar ge x expa nsions for these p olynomials in terms o f the a n and b n , g iving p n = γ n   x n − x n − 1 n − 1 X i =0 b i + x n − 2   n − 2 X i =0 n − 1 X j = i +1 b i b j − n − 1 X i =1 a 2 i   + O ( x n − 3 )   , (2.6a) ǫ n = γ − 1 n   x − n − 1 + x − n − 2 n X i =0 b i + x − n − 3   n X i =0 i X j =0 b i b j + n +1 X i =1 a 2 i   + O ( x − n − 4 )   , (2.6b) where in the se cond equation use has been made of the lar ge x expansion of γ n ǫ n = ( x − b n ) − 1 γ n − 1 ǫ n − 1 + ( x − b n ) − 1 a 2 n +1 γ n +1 ǫ n +1 [ 30, 17, 61 , 49 ]. Utilising this we can write some explicit rela tions b et w een the co eﬃcients, 6 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER γ n and γ n,k ’s, and the a n and b n , a n = γ n − 1 γ n , (2.7a) b n − 1 = γ n − 1 , 1 γ n − 1 − γ n, 1 γ n , (2.7b) which ho ld for n ≥ 1 . By equating ( f p n ) p n − 1 with ( f p n − 1 ) p n we hav e the rela tion (2.8) φ n − 1 p n − 1 − φ n − 2 p n = ǫ n − 1 p n − ǫ n p n − 1 = 1 a n , which is p olynomial by the left hand side and where the ﬁnal equality follows fr om (2.6). It is clear tha t the seq uence of functions , { ǫ n } ∞ n =0 , is a solution of (2.2) that is independent of { p n } . W e ﬁnd it co n venient to int ro duce the matr ix (2.9) Y n =   p n ǫ n /w p n − 1 ǫ n − 1 /w   . The thr ee term recursio n rela tion is eq uiv a len t to the r elation (2.10) Y n +1 = M n Y n , where M n =    ( x − b n ) a n +1 − a n a n +1 1 0    . 3. Connectio n pres erving deformations In this section we revis e the established classica l theor y of sys tems of linear q -diﬀerence eq ua tions [ 1, 10, 1 4 ]. The g eneral theory concerns the m × m matrix system (3.1) Y ( q x ) = A ( x ) Y ( x ) , where A ( x ) is rational in x . W e call A the co e ﬃcien t matrix of the linear q -diﬀerence equation. One may easily verify that such an equation p ossesses tw o symbolic solutions, namely , the inﬁnite pro ducts A ( x/q ) A ( x/q 2 ) A ( x/q 3 ) . . . , (3.2a) A ( x ) − 1 A ( xq ) − 1 A ( xq 2 ) − 1 . . . , (3.2b) which do not conv erge in genera l. W e may suitably trans form the pro blem so that A ( x ) is p olynomia l, which we parameterize by w r iting A ( x ) = A 0 + A 1 x + . . . + A n x n . The matrices A 0 and A n are assumed to be s e misimple with eigen v alues ρ 1 , . . . , ρ m and λ 1 , . . . , λ m resp ectively . Regarding the solutio ns of (3.1), we pr esen t the following theor em due to Ca rmichael [ 14 ]. Theorem 3. 1. Supp ose the eigenvalues of A 0 and A n satisfy t he c ondition ρ i ρ j , λ i λ j / ∈ { q , q 2 , q 3 , . . . } , CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 7 then t her e exists two solutions, c al le d t he fundamental solut ions, of the form Y 0 = b Y 0 x D 0 , Y ∞ = b Y ∞ q nu ( u − 1) 2 x D ∞ , wher e b Y 0 and b Y ∞ ar e holo morphic functions in n eighb ourho o ds of x = 0 and x = ∞ r esp e ctively and D 0 and D ∞ ar e diag(log q ρ i ) and diag(lo g q λ i ) r esp e ctively and u = log q x . W e may use (3.1) to co n tin ue b oth solutions meromorphically ov er C \ 0. F r om these solutions, we deﬁne the connection matrix to b e (3.3) P ( x ) = Y ∞ ( x ) − 1 Y 0 ( x ) . The e volution o f P 9 x ) in x is given by P ( q x ) = Y ∞ ( q x ) − 1 Y 0 ( q x ) , = Y ∞ ( x ) − 1 A ( x ) − 1 A ( x ) Y 0 ( x ) , = P ( x ) . (3.4) Hence this function is q -p erio dic in x . These fundamental solutions ma y b e rela ted to (3.2) via a conjugatio n o f transformations of (3.1) suc h that the solutions given by (3.2) conv erge. Hence (3.2) giv es us infor ma tion reg arding the r o o ts of the deter minan t of the connection matrix. If the z i are the zero s of det A ( x ), then Y − 1 ∞ and Y 0 are p ossibly singula r a t { q n +1 z i : n ∈ N } and { q − n z i : n ∈ N } resp ectively . Therefore we exp ect the p oles and zeros of the deter minan t of P ( x ) to be q -p ow e r m ultiples o f the z i . In the situation of mono dromy pres erving deformations w e introduce a pa rameter, t , in to A and consider what co nditions on the evolution of t are req uired so tha t the mono dromy repres en tation is preserved. It was the innov ation of J imbo and Sak a i [ 40 ] to introduce a para meter t in a manner that preser v es the connection ma trix, P ( x ), through the evolution t → q t . By the o bserv ation noted in the ab ov e paragraph regarding the p oles and ro ots o f the determinant of P ( x ), we may infer that the latter ar e preser v ed in the s hift z j → q z j . Ho wever, in doing this, we must also c o nsider the eigenv alues o f A 0 and A n to b e parameter s . This sug g ests tha t the zeros of the deter minan t o f A and eigenv a lue s of A 0 and A n are appropria te c hoices of para meter for the co nnection matrix preserving de fo rmation. W e parameterize ρ A = det A b y letting a subset of the ro ots b e c o nstant in t , while the o ther ro ots are simply prop ortional to t . By s upposing P ( x, t ) = P ( x, q t ) we arr iv e a t the implica tio n P ( x, q t ) = Y ∞ ( x, q t ) − 1 Y 0 ( x, q t ) = Y ∞ ( x, t ) − 1 Y 0 ( x, t ) = P ( x, t ) , which de ﬁnes a matrix, B , via Y 0 ( x, q t ) Y 0 ( x, t ) − 1 = Y ∞ ( x, q t ) Y ∞ ( x, t ) − 1 = B ( x, t ) . 8 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER Since b oth Y 0 and Y ∞ are independent solutions this leads to the necessa ry condition that the evolution of Y m ust be g o verned by a seco nd linear e q uation, (3.5) Y ( x, q t ) = B ( x, t ) Y ( x, t ) . Conv er sely , it is easy to see that if Y satisﬁe s (3.5), then the evolution t → q t deﬁnes a connection preser ving deformation. Since Y sa tisﬁes an eq uation in x a nd an equation in t this impo s es the necessar y compatibility condition (3.6) A ( x, q t ) B ( x, t ) = B ( q x, t ) A ( x, t ) . This compatibility co ndition implies a q -diﬀerence e q uation satisﬁed by ρ B = det B , ρ B ( q x, t ) = ρ A ( x, q t ) ρ A ( x, t ) ρ B ( x, t ) , which may b e solved up to a factor of a function of t . W e shall assume that ρ B is rationa l in x . Other informa- tion regarding asymptotic behavior of A gives us sp eciﬁc informatio n regarding the form of B . F urthermore the compatibility condition and the determinant al constraints often results in an ov er determined system allowing us to co ns truct a representation of B in terms o f the entries o f A and hence, ﬁnd q -diﬀerence eq uations in t for the ent ries o f A . W e do not pursue this line explicitly . Rather, in the following sections we will show ho w the q -diﬀerence equations satisﬁed b y a particular defo rmed q - semiclassical o r thogonal p olynomial system lea ds linear systems satisfying (3 .1) and (3.5). 4. q -di ﬀ erence equations satisﬁed by orthogonal p o l ynomials 4.1. The q -diﬀerence calculus and q -sp ecial functions. A r eference for the q - diﬀerence ca lculus and a lso the h -diﬀer en tial ca lculus is a bo ok by Kac [ 41 ]. W e ﬁrst r ecall s ome of the basic pr oper ties in r elation to q - diﬀerence equations. The ﬁrs t pr ope r t y is the q -a nalog of the pro duct and quotient r ule, given b y D q,x ( f g ) = f D q,x g + g D q,x f , (4.1a) = f D q,x g + g D q,x f , D q,x  f g  = g D q,x f − f D q,x g g g , (4.1b) where f = f ( x ) = f ( q x ). Asso ciated with the q -diﬀer e nce ope rator is the a n tideriv ative, known as Jackson’s q -integral, as deﬁned by Thomae and Jackson [ 36, 62, 63 ]. W e expres s the deﬁnite int egral o f Thomae [ 62, 63 ] by (4.2) Z 1 0 f ( x )d q x = (1 − q ) ∞ X k =0 q k f ( q k ) . This was subs equen tly genera lized by Jackson [ 36 ] to (4.3) Z b a f ( x )d q x = Z b 0 f ( x )d q x − Z a 0 f ( x )d q x, where Z a 0 f ( x )d q x = a (1 − q ) ∞ X k =0 q k f ( aq k ) . CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 9 If f is contin uous, then lim q → 1 Z a 0 f ( t )d q t = Z a 0 f ( t )d t. W e in tro duce the q -Po chh ammer s ym bol ( a ; q ) n = (1 − a )(1 − aq ) . . . (1 − a q n − 1 ) , and ( a ; q ) ∞ = (1 − a )(1 − aq ) . . . . W e also a dopt the notation ( a 1 , a 2 , . . . , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n . . . ( a m ; q ) n . The s ym bol ( a ; q ) ∞ is often called the q - expo ne ntial as D q,x ( ax ; q ) ∞ = − a ( aq x ; q ) ∞ (1 − q ) . W e are now able to express Heine’s ba sic hypergeo metric function [ 29 ], as it was re - written by Tho ma e [ 62, 26 ], (4.4) 2 φ 1   a, b c       q ; t   = ∞ X m =0 t m ( a, b ; q ) m ( c, q ; q ) m . Also relev ant is the integral formula fo r the basic hypergeo metric function [ 26 ] (4.5) 2 φ 1   a, b c       q ; t   =  b, c b ; q  ∞ (1 − q )( c, q ; q ) ∞ Z 1 0 x log b log q − 1 ( xta, xq ; q ) ∞  xt, xc b ; q  ∞ d q x. Jacobi’s elliptic multiplicativ e theta function, as deﬁned by J acobi’s triple pro duct for m ula, may be expressed as θ q ( z ) =  q , − q z , − 1 z ; q  ∞ . This satisﬁes the equa tion θ q ( q z ) = q z θ q ( z ) . Of imp orta nc e is the q -character, e q,c ( x ) = θ q ( x ) θ q (1 /c ) θ q ( x/c ) , satisfying (4.6) e q,c ( q x ) = ce q,c ( x ) , e q,q c ( x ) = xe q,c ( x ) . 10 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER 4.2. q -diﬀ erence e quation i n x . W e shall henceforth assume that the log q -deriv ative of the weigh t sp ecifying the linea r form of the q -o rthogonal p olynomial sy stem is rational, pa r ameterizing its q -deriv a tiv e via the equation (4.7) W ( x ) D q,x w ( x ) = 2 V ( x ) w ( x ) , where W ( x ) and V ( x ) a re p olyno mia ls in x . This is the q -ana logue of the no tion of semi-c la ssical weight functions familiar in the theor y of classical or thogonal p olynomia ls [ 50 ] and which was pro pos ed by Magnus[ 48 ] and subse - quently by others [ 51, 43 ]. Hence fo rth we r egard systems satisfying these conditions as q -semi- classical ortho gonal po lynomial sys tems . Lemma 4. 1. Assuming w satisﬁes (4.7) , the S tieltjes function, f , satisﬁes the q -diﬀer enc e e quation (4.8) W ( x ) D q,x f ( x ) = 2 V ( x ) f ( x ) + U ( x ) , wher e U is p olynomial su ch that deg U < deg V . W e call the p olynomials W , V and U the spectr al data polyno mials. W e will use (4.8) as the basis for our deriv ation of the x -ev olution for the q -semi-cla ssical orthog onal sys tems. Using diﬀerent metho ds, this has b een derived in [ 33, 16, 31, 3 2 ] for general (i.e. b eyond semi-clas s ical) q - orthogonal p olynomials. Theorem 4. 2. [ 48, 42 ] The matrix Y n satisﬁes (4.9) D q,x Y n = A n Y n , wher e A n = 1 ( W ( x ) − 2 x (1 − q ) V ( x ))   Ω n − V − a n Θ n a n Θ n − 1 Ω n − 1 − V − ( x − b n − 1 )Θ n − 1   , (4.10) and Θ n and Ω n ar e p olynomials sp e ciﬁe d by Θ n = W ( ǫ n D q,x p n − p n D q,x ǫ n ) + 2 V ǫ n p n , (4.11a) Ω n = a n W ( ǫ n − 1 D q,x p n − p n − 1 D q,x ǫ n ) (4.11b) + a n V ( p n ǫ n − 1 + p n − 1 ǫ n ) − 2 V x (1 − q ) a n ǫ n − 1 D q,x p n . Proof. Using (4.8) and f = φ n − 1 p n + ǫ n p n , we ﬁnd W D q,x  φ n − 1 p n  − 2 V φ n − 1 p n − U = − W D q,x  ǫ n p n  + 2 V ǫ n p n . This may b e expanded to g iv e W ( p n D q,x φ n − 1 − φ n − 1 D q,x p n ) − 2 V φ n − 1 p n − U p n p n p n p n = W ( ǫ n D q,x p n − p n D q,x ǫ n ) + 2 V ǫ n p n p n p n . Setting this equa l to Θ n / ( p n p n ) deﬁnes Θ n as Θ n = W ( p n D q,x φ n − 1 − φ n − 1 D q,x p n ) − 2 V φ n − 1 p n − U p n p n , (4.12) = W ( ǫ n D q,x p n − p n D q,x ǫ n ) + 2 V ǫ n p n . CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 11 The ﬁrst expression is a linear c o m bination of p olynomia ls whic h veriﬁes that Θ n is a polynomia l for all n . The second expression is one in whic h every o ccurre nce of a p n or its deriv a tiv e is mu ltiplied b y a factor of ǫ n or its deriv ative. Hence, by (2.6), there exists a n upp er b ound for the degr ee of the polynomia l Θ n in x that is indep endent of n . Using the fact that Θ n is po lynomial and (4.13) a n φ n − 1 p n − 1 − a n φ n − 2 p n = 1 , we hav e ( a n φ n − 1 p n − 1 − a n φ n − 2 p n )Θ n = W ( p n D q,x φ n − 1 − φ n − 1 D q,x p n ) − 2 V φ n − 1 p n − U p n p n . By appr opriately equating fa c tors divisible by φ n − 1 on one side and factor s divisible by p n on the other s ide, we let Ω n be the common factor by wr iting p n φ n − 1 Ω n = φ n − 1 ( a n Θ n p n − 1 + ( W − 2 V x (1 − q )) D q,x p n + V p n ) , = p n ( a n φ n − 2 Θ n + W D q,x φ n − 1 − V φ n − 1 − U p n ) . The ﬁrst e q ualit y is a rear rangement of the r equired q -diﬀerence equation for p n . The s econd ex pression for Ω n is equiv alent to (4.14) Ω n = a n φ n − 2 Θ n φ n − 1 + W D q,x φ n − 1 φ n − 1 − V − U p n φ n − 1 . W e use the expr ession for Θ n in ter ms of φ n − 1 and p n to give Ω n = W a n φ n − 2 p n D q,x φ n − 1 φ n − 1 − W a n φ n − 2 φ n − 1 D q,x p n φ n − 1 − 2 V a n φ n − 2 φ n − 1 p n φ n − 1 − U a n φ n − 2 p n p n φ n − 1 + W D q,x φ n − 1 φ n − 1 − V − U p n φ n − 1 , which, by using the equality a n p n φ n − 2 = a n p n − 1 φ n − 1 − 1 to eliminate o ccurrences o f φ n − 2 , is equiv alent to Ω n = a n W ( p n − 1 D q,x φ n − 1 − φ n − 2 D q,x p n ) − V (2 a n φ n − 2 p n + 1) − U a n p n − 1 p n . This expresses Ω n as a linear combination of p olynomials and hence Ω n is a poly no mial. T o obtain the large x expansion, by dividing the ﬁrst expressio n for Ω n by φ n − 1 p n we hav e (4.15) Ω n = a n p n − 1 Θ n p n + W D q,x p n p n + V − 2 x (1 − q ) V D q,x p n p n . Using (4.11 a ), we ﬁnd that Ω n may b e written as Ω n = a n p n − 1 W ǫ n D q,x p n p n − a n W p n − 1 p n D q,x ǫ n p n + 2 a n p n − 1 V ǫ n p n p n + W D q,x p n p n + V − 2 x (1 − q ) V D q,x p n p n . Using (4.13) to c a ncel the factors of p n − 1 gives (4.11 b). A r e arrangement of (4.15) is ( W − 2 x (1 − q ) V ) D q,x p n = (Ω n − V ) p n − a n Θ n p n − 1 , where Ω n and Θ n are given by (4.11a) and (4.11b). Mapping n → n − 1 in this relation and expressing p n − 2 in terms of p n and p n − 1 gives ( W − 2 x (1 − q ) V ) D q,x p n − 1 = a n Θ n − 1 p n + (Ω n − 1 − V − ( x − b n − 1 )Θ n − 1 ) p n − 1 , 12 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER showing that the or thogonal p olyno mia ls form a column vector solution of (4.9). T o s e e that ǫ n /w s a tisﬁes the same q - diﬀerence equation, we consider W D q,x φ n − 1 + W D q,x ǫ n = W D q,x ( f p n ), which we refor m ulate a s W D q,x φ n − 1 + W D q,x ǫ n = W p n D q,x f + W f D q,x p n , =2 V f p n + U p n + f ((Ω n − V ) p n − a n Θ n p n − 1 + 2 V p n − 2 V p n ) , = [ U p n + (Ω n + V ) φ n − 1 − a n Θ n φ n − 2 ] + [(Ω n + V ) ǫ n − a n Θ n ǫ n − 1 ] . Hence by s ubtracting W D q,x φ n − 1 , deﬁned by (4.1 4), we ﬁnd W D q,x ǫ n = (Ω n + V ) ǫ n − a n Θ n ǫ n − 1 , and from this D q,x  ǫ n w  = wD q,x ǫ n − ǫ n D q,x w ww , = (Ω n + V ) ǫ n − a n Θ n ǫ n − 1 W − 2 V ǫ n W w  1 − 2 x (1 − q ) 2 V W  , = (Ω n − V ) ǫ n w − a n Θ n ǫ n − 1 w W − 2 x (1 − q ) V , while the shift n → n − 1 and (2.2) gives the compa tible evolution e q uation for ǫ n − 1 /w .  Since p n is of degree n and the leading o rder term of ǫ n ab out x = ∞ is x − n − 1 , we immediately obtain upp er bo unds for the degrees o f Θ n and Ω n [ 48 ] deg Θ n ≤ max(deg W − 1 , deg V − 2 , 0) , (4.16a) deg Ω n ≤ max(deg W, deg V − 1 , 0) . (4.16b) It follows that knowledge of W a nd V may b e used in conjunction with (2.6) to determine Ω n and Θ n in ter ms of sums and pro ducts of the a i and b i . W e remark that (4.9) can b e rewritten in a manner mor e familiar in the context of c onnection matrices where Y n is a solution o f the linear q -diﬀerence equation (4.17) Y n ( q x ) = ( I − x (1 − q ) A n ) Y n ( x ) = ˜ A n Y n ( x ) . As rev is ed in Section 3, under appro pr iate co nditions on ˜ A n established by Carmichael [ 14 ], or the more g eneral conditions of Adams [ 1 ], this equation p ermits tw o fundamental so lutio ns: Y ∞ ,n and Y 0 ,n . W e for m the connection matrix by using the fundamental solutions of (4.9) in (3.3) (4.18) P n ( x ) = ( Y ∞ ,n ( x )) − 1 Y 0 ,n ( x ) . Solutions of (4.9) also s atisfy (2.1 0 ) and so the tr ansformation n → n + 1 is a c o nnection preser v ing deformation. This also implies that the connection matrix is indep enden t of the co ns ecutiv e p olynomials chosen in the orthogonal system. Therefore the connection matr ix is an inv a riant of the or thogonal p olynomial system. CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 13 In further sp ecifying the evolution in x for Y n , we hav e a c o mpatibilit y relatio n b e t ween the evolution in x a nd the evolution in n . This compatibility relation, deﬁned by equating the tw o ways of ev aluating D q,x Y n +1 , na mely D q,x M n Y n = A n +1 M n Y n , results in the co ndition (4.19) M n ( q x ) A n ( x ) + D q,x M n ( x ) = A n +1 ( x ) M n ( x ) . The e ntries of the ﬁrst row ar e equiv alent to the re c urrence relations o f Magnus [ 4 8, 42 ] ( x − b n )(Ω n +1 − Ω n ) = W − x (1 − q )(Ω n + V ) − a 2 n Θ n − 1 + a 2 n +1 Θ n +1 , (4.20a) Ω n − 1 − Ω n +1 =( x − b n − 1 )Θ n − 1 − ( q x − b n )Θ n . (4.20b) Using the r elation (2.8) we ﬁnd (4.21) det Y n = p n ǫ n − 1 − p n − 1 ǫ n w = 1 a n w . F rom this we may deduce (4.22) det( I − x (1 − q ) A n ) = det Y n ( q x ) det Y n ( x ) = w ( x ) w ( q x ) = W ( x ) W ( x ) − 2 x (1 − q ) V ( x ) , or eq uiv a len tly (4.23) x (1 − q ) det A n − T r A n = 2 V W − 2 x (1 − q ) V , and these in turn imply the a dditional recur rence r elation (4.24) x (1 − q )  Ω 2 n − V 2 − a 2 n Θ n Θ n − 1  = (( x − b n − 1 )Θ n − 1 − Ω n − 1 − Ω n )( W − x (1 − q )(Ω n + V )) . The c o mpatibilit y b et w een the evolution in x and n for mo re genera l orthog o nal p olynomials has given rise to asso ciated linea r pro blems for discr ete Painlev ´ e equatio ns [ 53, 64 ]. Many of these as so ciated linear problems are diﬀerential-diﬀerence systems [ 34 ]. That is to say that the evolution in x is deﬁned by a diﬀerential equation, while the e v olution of n is discrete. The ﬁrst occur rence o f a dis crete Painlev ´ e equation in the literatur e is thought to hav e b een deduced in this manner [ 60 ]. The 2 × 2 linear problem derived for ortho gonal p olynomials is one in which the co eﬃcient matrix, ˜ A n ( x ), is rational. If we follow the theo ry of connection matrices, we apply a transfor mation that r elates the linea r problem in which ˜ A n is ra tional to another linear pr oblem in which the coe ﬃcie n t matrix is polynomia l. With r e spect to Birkhoﬀ theo ry and (3.1), the co eﬃcient matrix ob eys the pro por tionalit y cons traint det A ∝ W ( W − 2 x (1 − q ) V ) . That is to say that when r e ferring to the co nnection matrix for or tho gonal p olynomial sys tems, we do not disting uish betw een the r oo ts and the p oles o f the determinant o f the linear problem. 14 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER 4.3. q -diﬀ erence equation in t . W e now turn to the new directio n that we wish to present. It is at this po in t we let w ( x ) = w ( x, t ), and hence co nsider p olynomials whic h are b oth functions of x and t . F or functions f ( x, t ) w ith tw o indep enden t pa rameters we will a dopt the no tation f = f ( x, t ) = f ( q x, t ) , ˆ f = \ f ( x, t ) = f ( x, q t ) . W e distinguish a ba se ca se in whic h deg Θ n = 0 and deg Ω n = 1, co rresp onding to deg V = 1 and deg W = 2, as being completely solv able a nd a ca s e in which the co nnection ma tr ix is known [ 46 ]. How ever by suitably a djoining q -exp onential factors that dep end simply on t to the numerator or denominator of w ( x ) we in tro duce ro ots or p oles int o w ( q x ) /w ( x ). This has the eﬀect of increa sing up the degree of W ( x ) and V ( x ). F urthermo re it impose s a rational character on the log a rithmic q - deriv ative of w with res p ect to t : (4.25) R ( x, t ) D q,t w ( x, t ) = 2 S ( x, t ) w ( x, t ) , where R ( x, t ) and S ( x, t ) ar e p olynomials in x . These cannot b e arbitrar y p olynomia ls in x as there is an implied compatibility condition. T his arises b ecause there ar e tw o wa ys of ca lculating the mixed deriv atives of w , namely D q,x D q,t w ( x, t ) and D q,t D q,x w ( x, t ), the equa lit y of which imp oses the constr a in t (4.26) 2 ˆ V ˆ W 2 S R − 2 S R 2 V W = D q,x 2 S R − D q,t 2 V W . A co nsequence of (4.25) is the following compa nion result to Lemma 4.1. Lemma 4. 3. The Stieltjes function, f , satisﬁes the q -diﬀer enc e e quation (4.27) RD q,t f = 2 S f + T , wher e T ( x, t ) is p olynomial such that deg x T < deg x S . Another compatibility rela tion is implied by D q,t D q,x f = D q,x D q,t f in c onjunction with (4.8) and (4.2 7). This relation can b e stated as (4.28) 2 ˆ V ˆ W T R − 2 S R U W = D q,x T R − D q,t U W . When (4.26) and (4.28) are s atisﬁed a companio n res ult to Theorem 4.2 c an b e stated. Theorem 4. 4. The matrix Y n is a solution to (4.29) D q,t Y n = B n Y n , wher e B n = 1 ( R − 2 t (1 − q ) S )   Ψ n − S − a n Φ n a n Φ n − 1 Ψ n − 1 − S − ( x − b n − 1 )Φ n − 1   , (4.30) with Φ n and Ψ n p olynomials in x sp e ciﬁe d by Φ n = R ( ǫ n D q,t p n − p n D q,t ǫ n ) + 2 S ǫ n ˆ p n , (4.31a) Ψ n = a n R n ( ǫ n − 1 D q,t p n − p n − 1 D q,t ǫ n ) + S (2 a n ǫ n − 1 ˆ p n − 1) . (4.31b) CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 15 Proof. Our strategy is to adapt the pr oo f of Theorem 4.2 to the D q,t op erator. Using RD q,t f = 2 S f + T and f = φ n − 1 p n + ǫ n p n we hav e RD q,t f = RD q,t  φ n − 1 p n + ǫ n p n  = 2 S φ n − 1 + ǫ n p n + T . This suggests that we deﬁne Φ n = R ( p n D q,t φ n − 1 − φ n − 1 D q,t p n ) − 2 S φ n − 1 ˆ p n − T p n ˆ p n , (4.32) = R ( ǫ n D q,t p n − p n D q,t ǫ n ) + 2 S ǫ n ˆ p n . Expressio n (4.3 2) tells us Φ n is a p olynomial in x while (4.31a) implies a b ound on the degr ee. Using (4.31a) and (4.1 3) we a r rive at a n ( p n − 1 φ n − 1 − p n φ n − 2 )Φ n = R ( p n D q,t φ n − 1 − φ n − 1 D q,t p n ) − 2 S φ n − 1 ˆ p n − T p n ˆ p n . By splitting this expressio n into ter ms divisible by φ n − 1 and p n , we arr iv e at an equa lit y that deﬁnes Ψ n , given by φ n − 1 p n Ψ n = φ n − 1 ( a n p n − 1 Φ n + ( R − 2 t (1 − q ) S ) D q,t p n + S p n ) , = p n ( a n φ n − 2 Φ n + RD q,t φ n − 1 − S φ n − 1 − T ˆ p n ) . The ﬁrst line is just a rearr angement of the req uired q -diﬀer e nce equation, in t , for p n . The second e x pression is equiv alent to Ψ n = a n φ n − 2 Φ n φ n − 1 + RD q,t φ n − 1 φ n − 1 − S − T ˆ p n φ n − 1 , = Ra n φ n − 2 p n D q,t φ n − 1 φ n − 1 − Ra n φ n − 2 φ n − 1 D q,t p n φ n − 1 − 2 S a n φ n − 2 φ n − 1 ˆ p n φ n − 1 − T a n φ n − 2 p n ˆ p n φ n − 1 + RD q,t φ n − 1 φ n − 1 − S − T ˆ p n φ n − 1 , = a n R ( p n − 1 D q,t φ n − 1 − φ n − 2 D q,t p n ) − S (2 a n φ n − 2 ˆ p n − 1) − a n T p n − 1 ˆ p n . W e remark that this, b eing a linear combination of p olynomials, implies Ψ n is a p olynomial in x . Using (4.31a) in the ﬁr st expre s sion fo r Ψ n allows us to write Ψ n = Ra n p n − 1 ǫ n D q,t p n p n − Ra n p n − 1 D q,t ǫ n + 2 S a n p n − 1 ǫ n ˆ p n p n + RD q,t p n p n − 2 t (1 − q ) S D q,t p n p n + S, which up on noting a n p n − 1 ǫ n = a n p n ǫ n − 1 − 1 implies (4.3 1b). The working to date shows ( R − 2 t (1 − q ) S ) D q,t p n = (Ψ n − S ) p n − a n Φ n p n − 1 . Replacing n by n − 1 in this expr ession, then using (2.2) to express p n − 2 in terms of p n and p n − 1 , es tablishes that p n and p n − 1 form a co lumn vector solution o f (4 .29). The deriv a tion of the q -diﬀerence equa tion in t fo r ǫ n /w ma y also b e der iv ed in an analo gous manner to the pro of of Theore m 4.2, so we refra in from the giving the details .  16 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER W e note that D q,t do es not necessa rily alter the degree in x , hence, the uppe r b ounds for the degrees of Φ and Ψ are given by deg x Φ n ≤ max(deg x S − 1 , deg x R − 1 , 0) , (4.33a) deg x Ψ n ≤ max(deg x S, deg x R, 0) . (4.33b) F urther to this, we may use (2.6) to determine co eﬃcients in terms of the a i and b i . Equation (4.29) may b e rew r itten in the context of connection preserving deformations to r e a d (4.34) Y n ( x, q t ) = ( I − t (1 − q ) B n ( x, t )) Y n = ˜ B n ( x, t ) Y n ( x, t ) . W e use this r elation and (4.18) to deduce P n ( x, q t ) =( Y ∞ ,n ( x, q t )) − 1 Y 0 ,n ( x, q t ) , =( Y ∞ ,n ( x, t )) ˜ B n ( x, t ) − 1 ˜ B n ( x, t ) Y 0 ,n ( x, t ) , = P n ( x, t ) , which s ho ws us that the co nnec tio n is pres erved under deformations in t . Since Y n satisﬁes (2.10) and (4.2 9) we ha ve a compatibility co ndition, which follows from a consideration of D q,t Y n +1 , (4.35) M n ( x, q t ) B n ( x, t ) + D q,t M n ( x, t ) = B n +1 ( x, t ) M n ( x, t ) . The ﬁr st row of (4.35) is equiv a le n t to x − b n a n +1 [ R − (1 − q ) t ( S + Ψ n +1 )] + (1 − q ) ta n +1 Φ n +1 = (4.36a) x − ˆ b n ˆ a n +1 [ R − (1 − q ) t ( S + Ψ n )] + (1 − q ) ta n ˆ a n Φ n − 1 ˆ a n +1 , a n ˆ a n +1 h (1 − q ) t ( x − ˆ b n )Φ n i + a n a n +1 [ R − (1 − q ) t ( S + Ψ n +1 )] = (4.36b) ˆ a n ˆ a n +1 [ R − (1 − q ) t ( S + Ψ n − 1 − ( x − b n − 1 )Φ n − 1 )] . W e hav e an additio na l relation (4.37) det( I − t (1 − q ) B n ) = det ˆ Y n det Y n = a n w ˆ a n ˆ w = a n R ˆ a n ( R − 2 t (1 − q ) S ) . A co nsequence of this r elation is the ﬁrst o rder recurr e nce re la tion in n , given by ˆ a n (( q − 1) t (Ψ n + S ) + R ) (( q − 1) t (Φ n − 1 ( b n − 1 − x ) + Ψ n − 1 + S ) + R ) (4.38) +( q − 1) 2 t 2 a 2 n Φ n − 1 Φ n ˆ a n − R a n (2( q − 1 ) S t + R ) = 0 . The c o mpatibilit y condition (4.19) is na turally paired w ith (4.3 5). Thus, re w r iting these r ead ˜ A n +1 ( x, t ) M n ( x, t ) = M n ( q x, t ) ˜ A n ( x, t ) , (4.39a) ˜ B n +1 ( x, t ) M n ( x, t ) = M n ( x, q t ) ˜ B n ( x, t ) . (4.39b) CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 17 A further identit y which can be gr ouped with these follows from the compatibilit y impo sed b y the requirement that D q,t D q,x Y n = D q,x D q,t Y n . One computes (4.39c) ˜ A n ( x, q t ) ˜ B n ( x, t ) = ˜ B n ( q x, t ) ˜ A n ( x, t ) , which is equiv alent to (3.6). 5. Deformed little q -Jacobi Polynomials The little q -Jacobi polyno mials were in tro duced by Hahn [ 28 ]. This family of po lynomials pos sesses the o r - thogonality relation [ 44 ] Z 1 0 x σ ( q xb ; q ) ∞ ( q x ; q ) ∞ p i ( x ) p j ( x )d q x = δ ij . This ratio of tw o exp onential factors may be sca led and chosen appropr iately so that the r o ot and p ole is a t a 3 and a 4 resp ectively . W e now adjoin tw o ro ots that are pro por tional to t , a 1 t and a 2 t , to give the deformed weight (5.1) w ( x, t ) = x σ  x a 1 t , x a 3 ; q  ∞  x ta 2 , x a 4 ; q  ∞ . In keeping with the notation of [ 40 ], we trust there is no ambiguit y b et ween the terms in the three ter m recurs io n relation, a n , a nd the r oo ts of the determinant, a i . The deformed p olynomials asso ciated with (5.1) satisfy L ( p i p j ) = Z S w ( x, t ) p i ( x, t ) p j ( x, t ) d q x = δ ij . The set S , also called the supp ort of the weigh t, may b egin and end at distinct roo ts of w ( x, t ). Thes e include a 3 , a 1 t a nd 0 . Cho osing a 3 and a 1 t and using (4.5) allows the moments to b e expressed in terms of Heine’s basic hypergeometr ic function, µ k = Z qa 1 t qa 3 x σ + k  x a 1 t , x a 3 ; q  ∞  x ta 2 , x a 4 ; q  ∞ d q x, = ( q a 1 t ) σ + k +1 (1 − q )  a 1 q σ + k + 2 a 2 , q ; q  ∞  q σ + k +1 , qa 1 a 2 ; q  ∞ 2 φ 1   a 4 a 3 , q σ + k +1 a 1 q σ + k + 2 a 2 q ; q a 1 t a 4   + ( q a 3 ) σ + k +1 (1 − q )  a 3 q σ + k + 2 a 4 , q ; q  ∞  q σ + k +1 , qa 3 a 4 ; q  ∞ 2 φ 1   a 2 a 1 , q σ + k +1 a 3 q σ + k + 2 a 4 q ; q a 3 a 2 t   . This allows us to use (2.5) to express a n and b n in terms of de ter minan ts of basic hypergeo metr ic functions. This weigh t (5.1) satisﬁes the equa tion D q,x w ( x, t ) =  a 2 a 4 ( x − a 1 t )( x − a 3 ) − q σ a 1 a 3 ( x − a 2 t )( x − a 4 ) a 2 a 4 ( x − a 1 t )( x − a 3 ) x (1 − q )  w ( x, t ) . A co mparison with (4.7) reveals that the sp ectral data p olynomials are W = a 2 a 4 ( x − a 1 t )( x − a 3 ) x (1 − q ) , 2 V = a 2 a 4 ( x − a 1 t )( x − a 3 ) − q σ a 1 a 3 ( x − a 2 t )( x − a 4 ) . 18 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER Recalling Theorem 4.2, it follows that the p oles o f the line a r q -diﬀerence equation in x sa tis ﬁe d b y these p olynomia ls is determined b y the p olynomial (5.2) W − 2 x (1 − q ) V = q σ a 1 a 3 ( x − a 2 t )( x − a 4 ) x (1 − q ) . In the t direction, w s atisﬁes the equation D q,t w ( x, t ) =  a 1 ( x − q a 2 t ) − a 2 ( x − q a 1 t ) a 1 ( x − q a 2 t ) t (1 − q )  w ( x, t ) . Comparing this expressio n with (4.25) shows R ( x, t ) = a 1 ( x − q a 2 t ) t (1 − q ) , 2 S ( x, t ) = a 1 ( x − q a 2 t ) − a 2 ( x − q a 1 t ) . The appro priate p o les o f the linea r q -diﬀerence eq uation in t satisﬁed by these p olynomials is therefor e determined by the p olynomia l (5.3) R − 2 t (1 − q ) S = t (1 − q )( x − q a 1 t ) . W e remark these explicit for ms for W , V , R and S s atisfy (4.26) as they must. 5.1. Linear problem. Since we hav e an upp er b ound fo r deg x Θ n , deg x Ω n , deg x Φ n and deg x Ψ n from (4.1 6) and (4.33), we para meterize Θ n , Ω n , Φ n and Ψ n by Θ n = θ 0 ,n + θ 1 ,n x, (5.4a) Ω n = ω 0 ,n + ω 1 ,n x + ω 2 ,n x 2 , (5.4b) Φ n = φ 0 ,n , (5.4c) Ψ n = ψ 0 ,n + ψ 1 ,n x. (5.4d) This b ounds the degree of the relev ant po ly nomial comp onent of the linear q -diﬀerence equatio ns in x and t . Hence the linea r q -diﬀerence equatio ns sa tisﬁed by the p olyno mials may b e written in the form (4.17) and (4.3 4) where ˜ A n = I − x (1 − q ) A n = ˜ A 0 ,n + ˜ A 1 ,n x + ˜ A 2 ,n x 2 ( x − a 2 t )( x − a 4 ) , (5.5) ˜ B n = I − t (1 − q ) B n = ˜ B 0 ,n + ˜ B 1 ,n x ( x − q a 1 t ) , (5.6) for some s et of ˜ A i,n and ˜ B i,n . According to (4.22) and (4.37), the determinants of these matrice s are det ˜ A n = a 2 a 4 ( x − a 1 t )( x − a 3 ) a 1 a 3 q σ ( x − a 2 t )( x − a 4 ) , det ˜ B n = a 1 a n ( x − a 2 q t ) a 2 ( x − a 1 q t ) ˆ a n . A t this po int, the asso ciated linear q -diﬀerence equation satisﬁed by the o rthogonal po ly nomials is one in whic h the co eﬃcient matrix, ˜ A n , is ratio nal r ather than polyno mial. T o rela te this formulation to the cla ssical theor y of Birkhoﬀ [ 10 ], or mor e pr ecisely , Jim bo and Sak ai [ 40 ], we require a gauge tra nsformation that will relate the linear q - diﬀerence eq uation in which the co eﬃcient ma tr ix is rationa l to a linear q -diﬀer ence equation in w hich the CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 19 co eﬃcien t matrix is poly nomial. By considering the as s ocia ted q -diﬀerence eq uation for Y ∗ n = Z n Y n , we no te that Y ∗ n satisﬁes the trio of equations Y ∗ n ( q x, t ) =  Z n ( I − x (1 − q ) A n ) Z − 1 n  Y ∗ n = A ∗ n Y ∗ n , (5.7a) Y ∗ n ( x, q t ) =  ˆ Z n ( I − t (1 − q ) B n ) Z − 1 n  Y ∗ n = B ∗ n Y ∗ n , (5.7b) Y ∗ n +1 ( x, t ) =  Z n +1 M n Z − 1 n  Y ∗ n = M ∗ n Y ∗ n . (5.7c) By letting Z n to b e prop ortional to a ppropriate q -exp onential factors allows A ∗ n to b e poly no mial. W e may also choose Z n carefully s o that A ∗ n po ssesses some desirable prop erties, such as ce r tain asymptotic characteristics in x and/or t , a nd doing so ma k es the co rresp ondence to the work o f Jimbo a nd Sak ai [ 40 ] mor e appa rent . Sp eciﬁcally , by choosing Z n ( x, t ) = e q,a 1 a 2 a 3 a 4 tq σ ( x )  x a 2 t , x a 4 ; q  ∞     a 2 a 4 q − n e q,q a 2 ( t ) γ n 0 0 γ n − 1 e q,q a 1 ( t )     , we hav e that Y ∗ n satisﬁes the q -diﬀer e nce equations Y ∗ n ( q x, t ) =( A ∗ 0 ,n + A ∗ 1 ,n x + A ∗ 2 ,n x 2 ) Y ∗ n = A ∗ n Y ∗ n , (5.8a) Y ∗ n ( x, q t ) = x ( B ∗ 0 ,n + B ∗ 1 ,n x ) ( x − q a 1 t )( x − q a 2 t ) Y ∗ n = B ∗ n Y ∗ n . (5.8b) The c o rresp onding determinants are given by (4 .22), (4.37) a nd (4.6) det( A ∗ n ) = a 1 a 2 a 3 a 4 q σ ( x − a 1 t )( x − a 2 t )( x − a 3 )( x − a 4 ) , (5.9a) det( B ∗ n ) = t 2 x 2 ( x − q ta 1 )( x − q ta 2 ) . (5.9b) It will transpire that the form of the coeﬃcie nt matrices of (5.8 a) a nd (5.8b) is w ell suited for the purp ose of parameterizing the linea r problem satisﬁed by the or thogonal po lynomials. Although (5.8a) sp eciﬁes a 2 × 2 linear q -diﬀerence sys tem in which the determinant of co eﬃcient ma tr ix, given by (5.9a), has ro ots that c oincide with those found in [ 40 ], we require tw o additiona l prop erties; ﬁrstly that A ∗ 2 ,n is a constant diago nal matrix and seco ndly , that A ∗ 0 ,n is semisimple with eigenv a lues prop ortiona l to t . An as ymptotic expansion of Ω n and Θ n around x = ∞ reveals ω 2 ,n = a 2 a 4 − q σ (2 q n − 1) a 1 a 3 2 , θ 1 ,n = a 2 a 4 q n +1 − q n + σ a 1 a 3 , giving (5.10) A ∗ 2 ,n =   κ 1 0 0 κ 2   , where κ 1 = q n + σ a 1 a 3 , κ 2 = a 2 a 4 q − n . This shows that the linear problem p osses s es the ﬁrst re quired pr oper t y . 20 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER T o s how that A ∗ 0 ,n has eig en v alues that are pr opo rtional to t , we ﬁrst wr ite A ∗ 0 ,n as A ∗ 0 ,n =    κ 1 κ 2 t + a 1 a 2 a 3 a 4 t 2 − ω 0 ,n q κ 2 w n θ 0 ,n κ 2 − q κ 1 a 2 n ( q κ 1 − κ 2 ) θ 0 ,n − 1 q κ 2 w n tκ 1 κ 2 + a 1 a 2 a 3 a 4 t 2 − b n − 1 θ 0 ,n − 1 − ω 0 ,n − 1    , (5.11) where we have used the notation (5.12) w n = e q,q a 1 ( t )( κ 2 − q κ 1 ) q e q,q a 2 ( t ) γ 2 n . W e let λ 1 ,n and λ 2 ,n be the eigenv alues of A ∗ 0 ,n . Utilizing (5.9) and that det( A ∗ n (0 , t )) = λ 1 ,n λ 2 ,n gives us λ 1 ,n λ 2 ,n = κ 1 κ 2 a 1 a 2 a 3 a 4 t 2 , revealing that ˆ λ 1 ,n ˆ λ 2 ,n = q 2 λ 1 ,n λ 2 ,n . Adding B ∗ 0 ,n to both sides of the r esidue of (4.39c) at x = 0, namely the relation ˆ A ∗ 0 ,n B ∗ 0 ,n = q B ∗ 0 ,n A ∗ 0 ,n , and then tak ing determinants shows det( I + ˆ A ∗ 0 ,n ) = det( I + q A ∗ 0 ,n ) , revealing 1 + ˆ λ 1 ,n + ˆ λ 2 ,n + ˆ λ 1 ,n ˆ λ 2 ,n = 1 + q λ 1 ,n + q λ 2 ,n + q 2 λ 1 ,n λ 2 ,n . This shows that λ 1 ,n λ 2 ,n ∝ t 2 and λ 1 ,n + λ 2 ,n ∝ t , hence λ 1 ,n and λ 2 ,n are pr opo rtional to t . A further pr oper t y o f λ 1 ,n and λ 2 ,n , that will b e useful la ter o n, is their indep endence o f n . The indep endence of κ 1 κ 2 ’s on n indicates that λ 1 ,n λ 2 ,n is independent of n . How ever the tra ce of A ∗ 0 ,n is κ 1 κ 2 t  1 + 1 q σ  − ( ω 0 ,n + ω 0 ,n − 1 + b n − 1 θ 0 ,n − 1 ) = λ 1 ,n + λ 2 ,n , which indica tes that the c o nstant co eﬃcient of (4.20) may b e expr essed in terms of λ 1 ,n and λ 2 ,n as λ 1 ,n + λ 2 ,n − λ 1 ,n +1 − λ 2 ,n +1 = 0 . This prov e s λ 1 ,n + λ 2 ,n is indep endent of n , hence λ 1 ,n and λ 2 ,n are indep endent of n . W e may now wr ite { λ 1 ,n , λ 2 ,n } = { θ 1 t, θ 2 t } , (5.13) where θ 1 and θ 2 are co nstant in t and n . These eigenv alues ar e not free, with an implicit dep endence on the a i ’s and κ i ’s and the supp ort chosen. The additional prop erties mean that (5.8a) ca n be cast in a for m e q uiv a len t to the linea r problem studied in [ 40 ]. T e c hnical achiev ement s in [ 40 ] are to identify the par ameterization o f the linear problem whic h leads to the q -P VI system. The present orthogo nal p olynomia l setting allows us to p erform these steps in a more detailed, and per haps more systematic manner. CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 21 5.2. Orthogonal p olynomial parameterization. Our pa th way tow ar d the par ameterization of the problem is to make use of the or thogonal p olynomial v a riables. Parameterizations of this so rt can be found in previous works [ 6, 8, 23, 24 ]. How e ver, these works do not pr o vide a systematic wa y to link up with co-ordinates that spec ify Painlev ´ e sys tems. T o b egin, using the e x pansions (4.11) of Ω n and Θ n , we ﬁnd ω 1 ,n = (1 − q ) κ 1 q n − 1 X i =0 b i + κ 1 ( a 2 t + a 4 ) − κ 1 κ 2 ( a 2 t + a 4 ) 2 a 2 a 4 − 1 2 a 2 a 4 ( a 1 t + a 3 ) , (5.14a) θ 0 ,n = κ 1  a 2 q t + a 4 q − q P n i =0 b i + P n − 1 i =0 b i  q − κ 2  a 1 q t + a 3 q + q P n − 1 i =0 b i − P n i =0 b i  q 2 , (5.14b) ω 0 ,n = a 2 tκ 1 n − 1 X i =0 b i ! + a 4 κ 1 n − 1 X i =0 b i ! − κ 1 a 2 t q n − 1 X i =0 b i ! − a 4 κ 1 q n − 1 X i =0 b i ! − κ 1 n X i =1 a 2 i ! + κ 1 q 2 n − 1 X i =1 a 2 i − κ 1 q 2 n − 2 X i =0 n − 1 X j = i +1 b i b j − κ 1 q 2 n − 1 X i =0 n − 1 X j = i b i b j + κ 1 q n − 1 X i =0 b i ! 2 + κ 2 q a 2 n − a 2 a 4 tκ 1 + a 1 a 2 a 3 a 4 t 2 + tκ 1 κ 2 2 . (5.14c) The e x pansions (4.31) of Φ n and Ψ n give ψ 1 ,n = 1 2 ( a 1 + a 2 ) − a 2 ˆ γ n γ n , (5.15a) ψ 0 ,n = a 2 ˆ γ n  P n − 1 i =0 ˆ b i − P n − 1 i =0 b i + a 1 q t  γ n − a 1 a 2 q t, (5.15b) φ 0 ,n = a 1 γ 2 n − a 2 ˆ γ 2 n γ n ˆ γ n . (5.15c) This sp eciﬁes a par a meterization of the linea r problem for Y ∗ n in terms of or thogonal p olynomial v ar iables. W e use the no ta tion (5.16) Γ n = n − 1 X i =0 b i , which is prop ortio nal to the co eﬃcient o f x n − 1 in p n . By combining (5.14), (5.4), (4.10) a nd (5.8a), (5.17) A ∗ 1 ,n =     ( q − 1 ) κ 1 Γ n q − κ 1 ( a 2 t + a 4 ) κ 2 w n a 2 n ( q κ 1 − κ 2 ) ( q κ 2 − κ 1 ) q 2 κ 2 w n − ( q − 1 ) κ 2 Γ n q − κ 2 ( a 1 t + a 3 )     . W e make use o f the relations trace A ∗ 0 ,n = θ 1 t + θ 2 t, det A ∗ 0 ,n = θ 1 θ 2 t 2 , which a llows (5.11) to be simpliﬁed to (5.18) A ∗ 0 ,n =     t ( a 1 a 2 a 3 a 4 + κ 1 κ 2 ) 2 − ω 0 ,n − q κ 2 w n θ 0 ,n q κ 1 − κ 2 ( q κ 1 − κ 2 ) a 2 n θ 0 ,n − 1 q κ 2 w n ω 0 ,n − t ( a 1 a 2 a 3 a 4 − 2 ( θ 1 + θ 2 ) + κ 1 κ 2 ) 2     , 22 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER where θ 0 ,n − 1 = ( a 1 a 2 a 3 a 4 t − 2 tθ 1 + tκ 1 κ 2 − 2 ω 0 ,n ) ( a 1 a 2 a 3 a 4 t − 2 tθ 2 + tκ 1 κ 2 − 2 ω 0 ,n ) 4 a 2 n θ 0 ,n . W e simplify the expressio n for A ∗ 0 ,n by introducing the v ariable r n so that we may wr ite A ∗ 0 ,n as (5.19) A ∗ 0 ,n =    θ 1 t + r n − q κ 2 w n θ 0 ,n q κ 1 − κ 2 ( q κ 1 − κ 2 ) r n ( r n + θ 1 t − θ 2 t ) q κ 2 w n θ 0 ,n θ 2 t − r n    . In r elating the co eﬃcient of x 2 in det A ∗ n with (5.9 a), we express r n as r n = − ( q − 1) κ 1 κ 2 ( a 1 t − a 2 t + a 3 − a 4 ) Γ n q ( κ 1 − κ 2 ) − (1 − q ) 2 κ 1 κ 2 Γ 2 n q 2 ( κ 1 − κ 2 ) − a 2 n ( q κ 1 − κ 2 ) ( q κ 2 − κ 1 ) q 2 ( κ 1 − κ 2 ) + t ( θ 2 κ 1 + θ 1 κ 2 − a 1 a 3 κ 2 κ 1 − a 2 a 4 κ 2 κ 1 ) κ 1 − κ 2 . Equating (5.1 9) with (5.18) gives a n a lternate representation of ω 0 ,n to that of (5.14 c). The equations (5.19), (5.17) and (5.10) are explicit para meterizations of the linear problem using orthogona l p olynomia l v a riables co m bined with knowledge o f the str uc tur es (5.8a) and (4.1 0). W e no w tur n our attention to the pa rameterization o f the linear problem, (5.8b), in volving B ∗ n . First, upon recalling (4.3 0), it follows fro m the large x expans ion of Φ n and Ψ n , a s implied by (5.15), tha t B ∗ 1 ,n = − tI . Direct substitution o f the v alues of Ψ n and Φ n from (5.15) gives (5.20) B ∗ 0 ,n = t     ˆ Γ n − Γ n + a 1 q t q κ 2 ( ˆ w n − w n ) q κ 1 − κ 2 ( q κ 1 − κ 2 )  w n ˆ a 2 n − a 2 n ˆ w n  q κ 2 w n ˆ w n Γ n − ˆ Γ n + q a 2 t     . This giv e s us enough infor mation to deduce the ev olution of the v ar iables γ 2 n , a 2 n and Γ n , which completes the parameteriza tion o f the linear pro blem in ter ms of the o rthogonal p olynomial v a riables. Use will b e made of (5.10), (5 .17), (5.1 9) and (5.20) as we now pro ceed to make the corresp ondence betw een the a bov e discre te dynamical s ystem and q -P VI by making a corres p ondence b etw een the linea r systems. 5.3. Jimbo - Sak ai parameterization. O ur primary task is to ﬁnd expr essions for w n , y n and z n in terms of γ 2 n , a 2 n and b n and vise versa. W e have c hosen w n in the previous para meterization, as it is rela ted to γ 2 n via (5.1 2), to be the v aria ble that reﬂects the g a uge freedom in b oth parameter izations of the linear pro ble m. In keeping with earlier remark s, we deduce (5.21) θ 0 ,n = y n ( q κ 1 − κ 2 ) q , and deﬁne v a riables z 1 and z 2 according to (5.22) A ∗ n ( y n , t ) =   κ 1 z 1 0 ∗ κ 2 z 2   . Ev aluating the determinant at x = y n reveals (5.23) z 1 z 2 = ( y n − a 1 t )( y n − a 2 t )( y n − a 3 )( y n − a 4 ) . CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 23 W e factorize this into the factors z 1 = ( y n − ta 1 )( y n − ta 2 ) q κ 1 z n , (5.24a) z 2 =( y n − a 3 )( y n − a 4 ) q κ 1 z n . (5.24b) The b eneﬁt of this pa rticular facto rization r e v eals itself in the pro of of Theore m 5 .1. It follows from (5.10), (5.11) and (5.19) tha t z 1 and z 2 may b e expressed in terms o f a 2 n and Γ n via the expre s sions κ 1 z 1 = − κ 1 θ 0 ,n (( q − 1)Γ n − q ( a 2 t + a 4 )) κ 2 − q κ 1 + q 2 κ 1 θ 2 0 ,n ( κ 2 − q κ 1 ) 2 + r n + tθ 1 , (5.25a) κ 2 z 2 = κ 2 θ 0 ,n (( q − 1)Γ n + a 1 q t + a 3 q ) κ 2 − q κ 1 + q 2 κ 2 θ 2 0 ,n ( κ 2 − q κ 1 ) 2 − r n + tθ 2 , (5.25b) which s peciﬁes z n . T o b e c o nsisten t with (5.22), the matr ix in (5.8a) p ermits the para meterization [ 40 ] (5.26) A ∗ n =    κ 1 (( x − y n )( x − α ) + z 1 ) κ 2 w n ( x − y n ) κ 1 ( γ x + δ ) w n κ 2 (( x − y n )( x − β ) + z 2 )    , where α , β , γ and δ are to b e deter mined. Comparing the upp er left entry o f (5.26) with (5.10), (5 .17) and (5 .19) shows Γ n = q ( a 2 t + a 4 − y n − α ) q − 1 , (5.27) r n = κ 1 y n α + κ 1 z 1 − tθ 1 . These substituted into (5 .25b) reveal (5.28) α = 1 κ 1 − κ 2  1 y n (( θ 1 + θ 2 ) t − κ 1 z 1 − κ 2 z 2 ) − κ 2 (( a 1 + a 2 ) t + a 3 + a 4 − 2 y n )  . Conv er sely , comparing co eﬃcients of the low er- left entry of (5.26) with (5 .10), (5.17) and (5.1 9) gives Γ n = − q ( a 1 t + a 3 − y n − β ) q − 1 , (5.29) r n = − κ 2 y n β − κ 2 z 2 + tθ 2 . These substituted into (5 .25a) show (5.30) β = 1 κ 1 − κ 2  − 1 y n (( θ 1 + θ 2 ) t − κ 1 z 1 − κ 2 z 2 ) + κ 1 (( a 1 + a 2 ) t + a 3 + a 4 − 2 y n )  . The strategy to b e used to sp ecify γ and δ ma k es use of (5.9 a). By equa ting the co eﬃcient of x 2 of det A ∗ n from (5.26) with (5.9 a) , we hav e (5.31) γ = z 1 + z 2 + ( y n + α )( y n + β ) + ( α + β ) y n − a 1 a 2 t 2 − ( a 1 + a 2 )( a 3 + a 4 ) t − a 3 a 4 . A co mparison b et w een the co eﬃcient of x in (5.26) with that o f (5.9a) shows (5.32) δ = 1 y n  a 1 a 2 a 3 a 4 t 2 − ( αy n + z 1 )( β y n + z 2 )  . This concludes our task of parameter izing the linear problem asso ciated with the orthogona l p olynomial system with the weigh t (5.1) and its corr espo ndence with the pa rameterization of [ 40 ]. After co mpleting the tas k of parameteriza tion o f α , β , γ and δ , Jimbo and Sak ai pr o ceeded to give the coupled equations referre d to as q -P VI . 24 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER How ever few details were given there. W e s hall pr o vide details by making use o f a structured form o f the B matrix following from the orthogonal p olynomial v ie wpoint. The structured for m of the B matrix follows by using the substitutions of (5.27) a nd (5.29) in (5.2 0), giving (5.33) B ∗ 0 ,n =     q t 2 ( a 1 + a 2 − D q,t ( y n + α )) − q tκ 2 ( w n − ˆ w n ) q κ 1 − κ 2 q tκ 1 ( ˆ w n γ − w n ˆ γ ) ( κ 1 − q κ 2 ) w n ˆ w n q t 2 ( a 1 + a 2 − D q,t ( y n + β ))     . In addition to (5.3 3) a crucial ingredient in our der iv ation o f q -P VI are the compatibility conditions (4.3 9). After making the tra nsformation Y ∗ n = Z n Y n these latter co nditions read A ∗ n ( x, q t ) B ∗ n ( x, t ) = B ∗ n ( q x, t ) A ∗ n ( x, t ) , (5.34a) A ∗ n +1 ( x, t ) M ∗ n ( x, t ) = M ∗ n ( q x, t ) A ∗ n ( x, t ) , (5.34b) B ∗ n +1 ( x, t ) M ∗ n ( x, t ) = M ∗ n ( x, q t ) B ∗ n ( x, t ) . (5.34c) By ev aluating the r esidue of (5.34a) at x = a 1 t, a 2 t, q a 1 t, q a 2 t , we obtain the ex pressions ( q a 1 tB ∗ 1 ,n + B ∗ 0 ,n ) A ∗ n ( a 1 t, t ) =0 , (5.35a) ( q a 2 tB ∗ 1 ,n + B ∗ 0 ,n ) A ∗ n ( a 2 t, t ) =0 , (5.35b) A ∗ n ( q a 1 t, q t )( q a 1 tB ∗ 1 ,n + B ∗ 0 ,n ) =0 , (5.36a) A ∗ n ( q a 2 t, q t )( q a 2 tB ∗ 1 ,n + B ∗ 0 ,n ) =0 . (5.36b) By lo oking at the residue o f (5.34a) at x = 0 , we o bta in the additional relation (5.37) ˆ A ∗ 0 ,n B ∗ 0 ,n = q B ∗ 0 ,n A ∗ 0 ,n . Theorem 5.1 ([ 40 ]) . The c omp atibility c ondition, (5.34a) , is e quivalent to the evolution e quations for y n , z n and w n sp e ciﬁe d by ˆ w n = w n ( q κ 1 ˆ z n − 1) κ 2 ˆ z n − 1 , (5.38a) ˆ z n z n = ( y n − a 1 t )( y n − ta 2 ) q κ 1 κ 2 ( y n − a 3 )( y n − a 4 ) , (5.38b) ˆ y n y n = q ( θ 1 ˆ z n − ta 1 a 2 )( θ 2 ˆ z n − ta 1 a 2 ) a 1 a 2 ( q κ 1 ˆ z n − 1)( κ 2 ˆ z n − 1) . (5.38c) Proof. F or brevity , we let the pa rameterization of B ∗ 0 ,n of (5.33) b e g iv en by B ∗ 0 ,n = ( b ij ) i,j =1 , 2 . The upp er right entries of compa tibility condition (5.36a) and (5.3 6 b) read κ 2 ˆ w n ( ˆ y n − q ta 1 )( b 22 − q a 1 t 2 ) = κ 1 b 12 (( q ta 1 − ˆ y n )( q ta 1 − ˆ α ) + ˆ z 1 ) , κ 2 ˆ w n ( ˆ y n − q ta 2 )( b 22 − q a 2 t 2 ) = κ 1 b 12 (( q ta 2 − ˆ y n )( q ta 2 − ˆ α ) + ˆ z 1 ) , which g iv es us an expr ession for b 12 and b 22 . W e only r equire b 12 , which is given by b 12 = tκ 2 ˆ w n ( ˆ y n − q a 1 t )( ˆ y n − q a 2 t ) κ 1 (( ˆ y n − q a 1 t )( ˆ y n − q a 2 t ) − ˆ z 1 ) . CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 25 Equating this with the upp er right element of (5.33) gives − q ( ˆ w n − w n ) q κ 1 − κ 2 = ˆ w n ( ˆ y n − q a 1 t )( ˆ y n − q a 2 t ) κ 1 ( ˆ z 1 − ( ˆ y n − q a 1 t )( ˆ y n − q a 2 t )) . This evolution eq uation is s impliﬁed using the pa rticular facto rization (5.23). The str ucture of the rig h t hand side of the a bove relation justiﬁes, a p osteri , the factorization (5 .23). The particula r form of (5.2 4) means the e volution of w n is equiv alent to (5.3 8a). The upp er right entries of compa tibilit y condition (5 .35a) and (5.35b) read κ 2 w n ( y n − a 1 t )( b 11 − q a 1 t 2 ) = κ 2 b 12 (( ta 1 − y n )( ta 1 − β ) + z 2 ) , κ 2 w n ( y n − a 2 t )( b 11 − q a 2 t 2 ) = κ 2 b 12 (( ta 2 − y n )( ta 2 − β ) + z 2 ) , which we solve in ter ms of b 11 and b 12 to give b 12 = q tw n ( y n − a 1 t )( y n − a 2 t ) ( y n − a 1 t )( y n − a 2 t ) − z 2 , (5.39) b 11 = q t ( z 2 ( y n − ( a 1 + a 2 ) t ) + β ( a 1 t − y n ) ( a 2 t − y n )) ( a 1 t − y n ) ( a 2 t − y n ) − z 2 . (5.40) W e deduce κ 2 ( ˆ w n − w n ) q κ 1 − κ 2 = w n ( y n − a 1 t )( y n − a 2 t ) ( y n − a 1 t )( y n − a 2 t ) − z 2 , which is equiv alent to (5.38b) knowing (5.38 a). Co mpa ring (5.40) with (5.33) yields t ( a 1 + a 2 + D q,t ( y n + α )) = z 2 ( y n − ( a 1 + a 2 ) t ) + β ( a 1 t − y n ) ( a 2 t − y n ) ( a 1 t − y n ) ( a 2 t − y n ) − z 2 , which is equiv alent to (5.38c) knowing (5.3 8 a) a nd (5.38b), o r the particular Ric c a ti solutio ns ˆ y n = q y n (1 − κ 2 ˆ z n ) 1 − q κ 2 ˆ z n , the la tter not b e ing sa tisﬁed in g eneral. The deriv ation of the evolution equa tions is complete.  F ull c orresp ondence with the J im b o and Sak ai fo r m is obta ine d by letting (5.41) a 5 = a 1 a 2 θ 1 , a 6 = a 1 a 2 θ 2 , a 7 = 1 q κ 1 , a 8 = 1 κ 2 , where (5.38) b ecome ˆ z n z n = a 7 a 8 ( y n − a 1 t )( y n − ta 2 ) ( y n − a 3 )( y n − a 4 ) , ˆ y n y n = a 3 a 4 ( ˆ z n − a 5 t )( ˆ z n − a 6 t ) ( ˆ z n − a 7 )( ˆ z n − a 8 ) , under conditions tha t a 5 a 6 a 7 a 8 = q a 1 a 2 a 3 a 4 , as g iven in [ 40 ]. 26 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER W e now return to the orthogo nal p olynomial context for these results. In addition to (5.7a) the three term recursion relation, (2.10), in the orthogonal polynomial context giv es us another linear problem. The representation of M ∗ n following from (2.1 0) and (5.7c) is (5.42) M ∗ n =    x − b n q κ 2 w n q κ 1 − κ 2 κ 2 − q κ 1 q κ 2 w n 0    . This ca n b e used to expr ess the orthogo nal p olynomial qua n tit y b n in terms of the na tur al v aria ble s . Co ns idering the co eﬃcient of x 2 and x in the upp er left and right entries of (5.34b) resp ectively r esults in the expression (5.43) b n = q ( q κ 1 α − κ 2 β ) q 2 κ 1 − κ 2 . F or the orthog onal p olyno mia l quantit y a 2 n a comparison of the low er left comp onent of A ∗ 1 ,n given by (5.17) and (5.26) s ho ws (5.44) a 2 n = q 2 κ 1 κ 2 γ ( q κ 1 − κ 2 )( q κ 2 − κ 1 ) . One imp ortant consequenc e from this per spective is that the natural v ar iables may b e expressed in terms of determinants of the moments. Using (5.12) and (2.5a) we have (5.45) w n = e q,q a 1 ( t )( κ 2 − q κ 1 )∆ n +1 q e q,q a 2 ( t )∆ n . Using (5.14 b) and (5.21) gives (5.46) y n = q κ 1 ( a 2 t + a 4 ) − κ 2 ( a 1 t + a 3 ) q κ 1 − κ 2 + κ 1 − κ 2 q κ 1 − κ 2 Σ n ∆ n − q κ 1 − κ 2 q q κ 1 − κ 2 Σ n +1 ∆ n +1 . The s imples t deter minan tal form fo r z n comes fro m the substitution o f (5.45) int o the inv ers ion of (5.38a), which reveals (5.47) z n = a 1 ∆ n +1 ∆ b n − a 2 ∆ n ∆ b n +1 a 1 κ 2 ∆ n +1 ∆ b n − q a 2 κ 1 ∆ n ∆ b n +1 . These may corresp ond to known de ter minan tal solutions, such as the Casorati determinants of Sak ai [ 56 ], a ltho ugh we are yet to inv estiga te this p oint. 5.4. B¨ ac kl und tra nsformations. T he linear problem equiv a le n t to the orthogo nal p olynomials three ter m recursion, (5 .7c), may b e expressed in terms of the natura l v a riables appear ing in (5.43). Substitution of (5.43) into (5.42) g iv es M ∗ n =    q 2 κ 1 ( x − α ) + κ 2 ( q β − x ) q 3 κ 1 − q κ 2 κ 2 w n q κ 1 − κ 2 κ 2 − q κ 1 q κ 2 w n 0    . In the cont ext of or thogonal p olynomial theory the system o f equations descr ibing the e volution of this sys tem in the n dir ection a r e k no wn the Lag uerre-F reud equations. Mor eov er, these very re currence relations in the tr a nsformation n → n − 1 and n → n + 1 repr e sen t elements in the group of B ¨ acklund transfor mations. Since the gr oup o f B¨ a c klund CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 27 transformatio ns ar e o f aﬃne W eyl t yp e, the L a guerre-F r eud equations are equiv a len t to a tra nslational comp onent of the extended aﬃne W eyl gro up of type D (1) 5 . W e repr esent the n → n − 1 translation as (5.48)    a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 : y n z n    →    a 1 a 2 a 3 a 4 a 5 a 6 q a 7 a 8 q : y n − 1 z n − 1    . The deriv ation of its explicit for m relies on (5 .34b). The low er right entry of (5.34 b) shifted n → n − 1 at x = y n − 1 yields the r elation (5.49) y n − 1 = − δ γ . By ev aluating the upp er right entry of (5.34 b) shifted by n → n − 1 at x = y n − 1 we obtain (5.50) z n − 1 = − ( y n − 1 − y n )( y n − 1 − α ) + z 1 q κ 2 ( a 4 − y n − 1 ) ( y n − 1 − a 3 ) . Finally , using (5.12) to ﬁnd w n − 1 /w n reveals w n − 1 = w n ( κ 1 − q κ 2 ) a 2 n ( q κ 1 − κ 2 ) , which e x presses y n − 1 , z n − 1 and w n − 1 in ter ms of y n , z n and w n . A more cano nical tra nsformation fro m the orthog onal p olynomial pers pective is the transforma tion cor resp ond- ing to the shift n → n + 1 , which is repr esent ed b y    a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 : y n z n    →    a 1 a 2 a 3 a 4 a 5 a 6 a 7 q q a 8 : y n +1 z n +1    . Another viewp oint is that this shift is a q - diﬀer ence analog ue of a Schlesinger transfor mation of the linea r system, which induces a B ¨ acklund transfor ma tion of the Painlev´ e equatio n[ 38 ]. The Schlesinger tra nsformation is induced by m ultiplicatio n on the le ft b y a ra tional matrix, this rational matrix coincides with M n ( x ) for this particular solution of the linear system. Theorem 5. 2. The shift ( y n , z n ) → ( y n +1 , z n +1 ) is given by (5.51) z n +1 = κ 2 z n [ y n ( a 1 t − y n ) + ζ n ] [ y n ( a 2 t − y n ) + ζ n ] q 2 κ 1 [ κ 2 y n z n ( a 3 − y n ) + ζ n ] [ κ 2 y n z n ( a 4 − y n ) + ζ n ] , y n +1 = κ 2 y n (1 − κ 2 z n ) q 2 κ 1 (1 − q 2 κ 1 z n +1 ) (5.52) ×     ζ n − ( y n − a 1 t )( y n − a 2 t ) + z n ( q θ 1 t − κ 2 a 1 a 2 t 2 ) 1 − κ 2 z n κ 2 y n z n ( a 3 − y n ) + ζ n         ζ n − ( y n − a 1 t )( y n − a 2 t ) + z n ( q θ 2 t − κ 2 a 1 a 2 t 2 ) 1 − κ 2 z n κ 2 y n z n ( a 4 − y n ) + ζ n     , wher e ( κ 2 − q 2 κ 1 ) ζ n = κ 2 ( y n − a 1 t )( y n − a 2 t ) − q 2 κ 1 κ 2 ( y n − a 3 )( y n − a 4 ) z n + κ 2 z n 1 − κ 2 z n ( tθ 1 − q κ 1 a 3 a 4 )( tθ 2 − q κ 1 a 3 a 4 ) κ 1 a 3 a 4 . 28 CHRISTOPHER M. ORMEROD, N. S. WITTE AND PETER J. FOR RESTER Proof. Using (5.34 b), we note that an a lternate wa y of w r iting A ∗ n +1 is given by (5.53) A ∗ n +1 ( x, t ) = M ∗ n ( q x, t ) A ∗ n ( x, t ) ( M ∗ n ( x, t )) − 1 . Using (5.26) to r epresent the top r ow, and the r igh t hand side o f (5.53) to e x press the b ottom r ow, we have A ∗ n +1 =    q κ 1 (( x − y n +1 )( x − ˜ α ) + ˜ z 1 ) q − 1 κ 2 w n +1 ( x − y n +1 ) − ( κ 2 − q κ 1 ) 2 ( x − y n ) q κ 2 w n ( x − y n )  b n  κ 1 − q − 1 κ 2  − κ 1 α + q − 1 xκ 2  + z 1 κ 1    , where ˜ z 1 and ˜ α denotes z 1 and α at n + 1. The determinan t of A ∗ n +1 at x = a 1 t is z ero. How ever, using this representation of A ∗ n +1 , the top row is divisible by ( y n +1 − a 1 t ) a nd the b ottom row is divisible by ( y n − a 1 t ). This also applies to the case for x = a 2 t , hence by equating the determinant of A ∗ n +1 with zer o g iv es tw o expressions for w n +1 −  y n +1 − a 2 t z n +1 − q 2 κ 1 ( a 1 t − ˜ α )   y n − a 2 t z n − ( q κ 1 − κ 2 ) b n + q κ 1 α − a 1 tκ 2  = ( κ 2 − q κ 1 ) 2 w n +1 w n , (5.54) −  y n +1 − a 1 t z n +1 − q 2 κ 1 ( a 2 t − ˜ α )   y n − a 1 t z n − ( q κ 1 − κ 2 ) b n + q κ 1 α − a 2 tκ 2  = ( κ 2 − q κ 1 ) 2 w n +1 w n . (5.55) In a similar manner, we consider the matrix repres en tation of A ∗ n +1 given by A ∗ n +1 =    ( x − y n )  κ 1 ( q x − b n ) + q − 1 κ 2 ( b n − q β )  + κ 2 z 2 q − 1 κ 2 w n +1 ( x − y n +1 ) − ( κ 2 − q κ 1 ) 2 ( x − y n ) q κ 2 w n κ 2 q (( x − y n +1 )( x − ˜ β ) + ˜ z 2 )    , which has b een obtained by using the left hand side of (5 .53) to represent the left c o lumn o f A ∗ n +1 and the r ight hand side of (5.53) to represent the right co lumn o f A ∗ n +1 . The left a nd right columns are divisible by y n − a 3 and y n +1 − a 3 resp ectively at x = a 3 . T his applies a lso in the case of x = a 4 . Hence equating the determinant of this representation of A ∗ n +1 at x = a 3 and x = a 4 with zer o g iv es tw o additiona l equations for w n +1 −  z n ( a 4 − y n ) + b n q  1 q κ 1 − 1 κ 2  − β q κ 1 + a 3 κ 2  z n +1 ( a 4 − y n +1 ) + a 3 − ˜ β q 2 κ 1 ! = 1 q 2  1 q κ 1 − 1 κ 2  2 w n +1 w n , (5.56) −  z n ( a 3 − y n ) + b n q  1 q κ 1 − 1 κ 2  − β q κ 1 + a 4 κ 2  z n +1 ( a 3 − y n +1 ) + a 4 − ˜ β q 2 κ 1 ! = 1 q 2  1 q κ 1 − 1 κ 2  2 w n +1 w n . (5.57) Equating the co eﬃcient of x in the upp er r ight entry of (5.34b) with zer o r ev eals ˜ α + y n +1 = ( q κ 1 − κ 2 ) b n + q ( q κ 1 y n + κ 2 β ) q 2 κ 1 , which reduces (5.54-5.57) to expressio ns for w n +1 that are a ll of deg r ee one in y n +1 and z n +1 . The compa tibility betw een (5.54) a nd (5.55) is e quiv a len t to y n +1 z n  κ 2 − q 2 κ 1   q 2 κ 1 z n +1 − 1  = a 3 a 4 κ 1 κ 2  κ 2 z n − q 2 κ 1 z n +1  ( a 1 a 2 t − q θ 1 z n ) ( a 1 a 2 t − q θ 2 z n ) θ 1 θ 2 y n ( κ 2 z n − 1) (5.58) − q 2 κ 1 ( z n − z n +1 ) (( a 3 + a 4 ) κ 2 z n − ( a 1 + a 2 ) t ) + q 2 y n κ 1 ( κ 2 z n − 1) ( z n − z n +1 ) . as is tha t o f (5.56 ) and (5 .57). Substituting (5 .58) into the equation resulting from the comparison of (5.54) and (5.56) the y ields (5.5 1). T o obtain (5.52), w e substitute the expressions for z n +1 , given b y (5.51), int o the rig h t hand side of (5.58).  CONNECTION PRESER VING DEF ORMA TIONS AND q -SEMI-CLASSICAL OR THOGONAL POL YNOMIALS 29 As a preliminary c he ck of the r ecurrence rela tions, we may consider the sp ecial case in whic h the suppor t is chosen to b e b et w een 0 and q a 1 t . In this sp ecial ca se the moments ar e µ k = ( q a 1 t ) σ + k +1 (1 − q )  a 1 q σ + k + 2 a 2 , q ; q  ∞  q σ + k +1 , qa 1 a 2 ; q  ∞ 2 φ 1   a 4 a 3 , q σ + k +1 a 1 q σ + k + 2 a 2 q ; q a 1 t a 4   . W e explicitly compute the eig en v alues of A ∗ 0 to b e θ 1 = q σ a 1 a 2 a 3 a 4 , θ 2 = a 1 a 2 a 3 a 4 . Substituting these v a lue s of µ k int o (5.46), (5 .47), (2.3) and (2.4 ) for the n = 0 case gives us the seed solution y 0 = a 2 a 4 ( a 1 t + a 3 ) − a 1 a 3 ( a 2 t + a 4 ) q σ +1 a 2 a 4 − a 1 a 3 q σ +1 − a 1 a 2 t  q σ +1 − 1   a 1 a 3 q σ +2 − a 2 a 4  2 φ 1   a 4 a 3 , q σ +2 a 1 q σ +3 a 2 q ; qa 1 t a 4   ( a 1 q σ +2 − a 2 ) ( a 1 a 3 q σ +1 − a 2 a 4 ) 2 φ 1   a 4 a 3 , q σ +1 a 1 q σ +2 a 2 q ; qa 1 t a 4   , z 0 = a 2 q − σ − 1 2 φ 1   a 4 a 3 , q σ +1 a 1 q σ +2 a 2 q ; a 1 t a 4   − a 1 2 φ 1   a 4 a 3 , q σ +1 a 1 q σ +2 a 2 q ; qa 1 t a 4   a 1 a 2 a 3 2 φ 1   a 4 a 3 , q σ +1 a 1 q σ +2 a 2 q ; a 1 t a 4   − a 1 a 2 a 4 2 φ 1   a 4 a 3 , q σ +1 a 1 q σ +2 a 2 q ; qa 1 t a 4   . As an illustration of the c omputation conten t of the recurrence relations and as a chec k on their veracit y , we may co mpare n umer ical v alue s of y n +1 and z n +1 using (5.4 6), (5.47), (2.3) and (2.4) found by us ing (5.52) and (5.51) fro m y n and z n for gener ic v alues o f the parameters , t a nd small v alues of n . Numerical evidence has be e n obtained to v e r ify that ( y 1 , z 1 ), found using (5.46), (5.4 7), (2.3) and (2.4), coincides with the v a lues of ( y 1 , z 1 ) found by using (5.52) a nd (5.51) from the v alues of ( y 0 , z 0 ) and (5.46), (5.4 7), (2.3) and (2.4). In a similar manner , w e were als o able to test the relationship b etw een ( y 1 , z 1 ) and ( y 2 , z 2 ) using (5.52) and (5.51) co mpared with v alues obtained by using (5.4 6), (5.47), (2.3) and (2 .4) . W e rema rk that the evolution n → n + 1 of the linea r system corresp onding to a deformed version of the Pastro weigh t s upported on the unit circle, which is the circular a nalogue of the little q -Jaco bi weigh t, ha s recently b een obtained by Bia ne [ 9 ]. The s tr ucture of the itera tions in n should hav e a s imilar structure to o ther tra nslational comp onen ts of the aﬃne W ey l gr oup, such as the translational component that coincides with the evolution of q -P VI . This multiplicativ e structure is of (5.52) and (5.51) is similar to the B¨ acklund trans formation of Biane [ 9 ]. In the work of Biane [ 9 ], the B¨ acklund transfor mation, r epresenting the shift n → n + 1, sim ultaneously changes one of the eig en v alues of A ∗ 0 ,n and A ∗ 2 ,n , wher eas in our tra nsformation, the eigenv a lues of A ∗ 0 ,n are indep endent o f n . T he little q -Jac obi cas e ha s als o b een studied in [ 27 ], although in a tr uncated w ay . 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