Semi-classical Laguerre polynomials and a third order discrete integrable equation

Nov ember 6, 2018 Semi-classical Laguerre p olynomials and a third order d iscrete in tegrable equation Paul E. Spicer Katholieke Un iversitei t L euven, Dep artement Wiskunde, Celestijnenlaan 200B B-3001 L euven, Belgium Email: P aul.Spicer @ wis.kuleuven.be and F rank W. Nijhoff Dep artment of Applie d Mathematics, U niversity of L e e ds, L e e ds LS2 9JT, U.K. Email: frank.nijhoff@gmail.co m Abstract A semi-discrete Lax pair formed from the differential system and r ecurrence r elation for s emi- classical orthogonal po lynomials, leads to a discr ete in tegr able equation for a sp ecific semi-classica l orthogo nal po ly nomial weigh t. The main exa mple we use is a semi-classica l Laguer re weigh t to derive a third order difference equa tion with a cor resp onding Lax pair. 1 In tro duction The connection b et ween semi-clas s ical or tho gonal p olynomials and discrete integrable systems is well establis he d. The ea r liest exa mple of a dis crete int egr able s ystem in semi-cla ssical or thogonal po lynomials can b e a ttributed first to Shohat in 193 9 [16], then second b y F r eud [10] in 197 6. How ever it wasn’t until the 1990’s, when the fo cus within in tegr a ble systems shifted from con tinuous to discrete, that F ok as, Its, Kitaev, [6] gave this equation a name; dis crete Painlev ´ e I, (d-P I ). Since then, other examples of discr ete Painlev ´ e equations have be e n found thro ugh explo ring the recur siv e structures of different se mi- classical orthogo nal p olynomial families, including semi- classical Hermite [15], semi-classic al Laguer re [9] and semi-cla ssical Charlier [1 7]. W e define a n orthogo nal p olynomial sequence { P n ( z ) } ∞ n =0 with r espect to a weight function 1 w ( z ) on an in terv al ( a, b ) as Z b a P n ( z ) P m ( z ) w ( z ) dz = h n δ nm , (1.1) with the corr e sponding r ecurrence r elation z P n ( z ) = P n +1 + S n P n + R n P n − 1 (1.2) for a mo nic orthogonal polynomial family P n ( z ) = z n + p nn − 1 z n − 1 + p nn − 2 z n − 2 + . . . F rom Bochner [2] we know that if { P n ( z ) } is a sequence of classic al orthog onal po lynomials, then P n ( x ) is a solution of the second-or der different ial equation φ ( z ) d 2 y dz 2 + ψ ( z ) dy dz = λ n y (1.3) where φ ( z ) a nd ψ ( z ) ar e fixed p olynomia ls of degree ≤ 2 and ≤ 1 resp ectively , a nd λ n is a rea l nu mber dep ending on the degree of the po lynomial solution. As a consequence of this the w eights of cla ssical or thogonal p olynomials satisfy a fir st orde r differential equation called the Pearson differential equation d dz ( φ ( z ) w ( z )) = ψ ( z ) w ( z ) , (1.4) when the degrees of φ and ψ satisfy deg φ ≤ 2 and deg ψ = 1 . How ever when the deg φ > 2 and \ o r deg ψ > 1 then the w eight function pro duces a class of semi-cla ssical or thogonal p olynomials. Our approa c h to semi-cla ssical orthog onal p olynomials is to make use of the Laguer re metho d [11] ( not to b e confused with La g uerre o rthogonal p olynomials ), which derives a pair of fir st o r der differential eq ua tions for a general class of orthogonal polyno mials, after the reduction of con tinued fractions. T he connection with semi-cla ssical or thogonal p olynomials o ccurs b ecause we asso ciate the s ystem with a semi-classica l weigh t function w ( x ) of the p olynomials with the Pearso n equation (1.4). F or co n venience we choo se to write the Pearso n e q uation in the following form W ( z ) ∂ z w ( z ) = V ( z ) w ( z ) , (1.5) where V ( z ) = ψ − φ ′ and W ( z ) = φ . While our a im and approa c h is different, the Laguerre metho d ha s been used to find co nnections with in tegr able systems, including contin uous Painlev ´ e equations, recently . Magnus [14], found a contin uous Painlev´ e equa tio n of the sixth k ind fr o m the r ecurrence c o efficients o f a se mi-classical J acobi p olynomial and F or rester a nd Witte [7, 8], found a Painlev´ e equation of the fifth kind, also using the Lag uerre metho d, but one that has b een extended to include bi-or tho gonal p olynomials. Our work will consider a semi-classical Laguerre weigh t, s imila r to that used by [9]. The semi- classical Lague r re p olynomials ha ve not b een as widely explored a s the semi-c la ssical Hermite 2 po lynomials, nor ar e they as complex as the semi-cla ssical Jacobi p olynomia ls. Thus, Laguerr e po lynomials ar e an appr o priate choice fo r finding new discr e te integrable systems. In section 2 we use the so-ca lled Laguerre metho d to derive the differential s ystem for semi- classical (monic) o rthogonal p olynomials. W e show how the compatibility b et ween the differential system a nd the recur rence relation (1.2 ) leads to a semi-discrete Lax equa tion [13], from which discrete in tegrable systems c a n b e deriv ed for sp ecific semi-classical orthogonal p olynomial weigh ts. In section 3 we choos e the semi-classica l L a guerre weigh t l 0 ( x ) = ( x − t ) α e − ( ax + b 2 x 2 ) which leads to a coupled difference system and a cor respo nding third or de r nonlinear difference eq ua tion. 2 The Laguerre Metho d W e introduce a moment g enerating function, the Stieltjes function, f ( z ) = Z w ( x ) z − x d ( x ) (2.1) (Stieltjes transform of the or thogonality measur e w ( x )) then equatio ns for P n can b e summarized as f ( z ) P n ( z ) = P (1) n − 1 ( z ) + ǫ n ( z ) , (2.2) where P (1) n − 1 ( z ) is an asso ciated poly nomial to P n ( z ), with degr ee n − 1. Although ǫ n ( z ) is not a po lynomial, w e can define it as ǫ n ( z ) = Z P n ( x ) z − x w ( x ) dx. (2.3) The p olynomials P (1) n − 1 ( z ), as well as the ǫ n ( z ), satisfy the sa me recurrence relations (1.2), but with P (1) − 1 ( z ) = 0. Additionally we have the following relations b etw een P n , P (1) n and ǫ n P n P (1) n − 2 − P n − 1 P (1) n − 1 = − h n − 1 (2.4a) P n − 1 ǫ n − P n ǫ n − 1 = − h n − 1 , (2.4b) which can b e found us ing the Chr istoffel-Darb oux iden tity . Since bo th P n ( z ) and ǫ n ( z ) sa tisfy the recurrence relation (1.2) we ca n give an explicit form of P n ( z ) and ǫ n ( z ) defined in terms of the recurrence relation’s co efficients: P n ( z ) = z n −   n − 1 X j =0 S j   z n − 1 + n − 1 X j =1 j − 1 X k =0 S j S k − R j ! z n − 2 + · · · (2.5a) ǫ n ( z ) = h n   1 z n +1 +   n X j =0 S j   1 z n +2 + n X j =0 R j +1 + j X i =0 S j S i ! 1 z n +3 + · · ·   . (2.5b) 3 Semi-classica l or thogonal p olynomials may be defined thr ough a differential difference equation of the form W ( z ) ∂ z f ( z ) = V ( z ) f ( z ) + U ( z ) (2.6) which comes from considering W ( z )( ∂ z f ( z )) and the Pearson equa tion (1.5). W ( z )( ∂ z f ( z )) = − Z W ( z ) w ( x ) ( z − x ) 2 dx = − Z d dx  1 z − x W ( z ) w ( x )  dx + Z W ( z ) z − x ∂ x w ( x ) = Z W ( z ) W ( x ) V ( x ) 1 z − x w ( x ) dx = V ( z ) f ( z ) + W ( z ) Z  V ( x ) W ( x ) − V ( z ) W ( z )  w ( x ) z − x dx On the fir st line we ass ume the first term reduce s to ze r o b e cause of par ameter co nstrain ts a nd then we hav e that: U ( z ) = W ( z ) Z  V ( x ) W ( x ) − V ( z ) W ( z )  w ( x ) z − x dx , where U ( z ) is a p olynomial in z . 2.1 The fundamen tal linear system for semi-classical ort hogonal p oly- nomials W e star t with the equation (2.2), differentiate it and m ultiply b y W , so tha t we can then make use of the firs t order linear differential equation (2.6) (with the ex c eption, that for this ca se we will consider the x v ariable to be dominan t). W f ∂ x P n + ( V f + U ) P n = W ( ∂ x P (1) n − 1 + ∂ x ǫ n ) W ∂ x P n ( P (1) n − 1 + ǫ n ) + V P n ( P (1) n − 1 + ǫ n ) + U P 2 n = W ( ∂ x P (1) n − 1 + ∂ x ǫ n ) P n (2.7) W e then go a bout separating the p olynomial ex pression P (1) n − 1 and ǫ n so w e get the following tw o equiv alen t expressio ns, which we deno te Θ n Θ n = W ( ∂ x P (1) n − 1 P n − ∂ x P n P (1) n − 1 ) − U P 2 n − V P n P (1) n − 1 , (2.8a) = W ( ∂ x P n ǫ n − ∂ x ǫ n P n ) + V P n ǫ n , (2.8b) where Θ n is a po lynomial b ounded by a co nstan t. W e try the same metho d aga in except this time we use f P n − 1 , which is ag ain differentiated a nd multiplied b y W . ∂ x f P n − 1 + f ∂ x P n − 1 = ∂ x P (1) n − 2 + ∂ x ǫ n − 1 (2.9) V P n − 1 ( P (1) n − 1 + ǫ n ) + U P n P n − 1 + W ∂ x P n − 1 ( P (1) n − 1 + ǫ n ) = W ( ∂ x P (1) n − 2 + ∂ x ǫ n − 1 ) P n (2.10) 4 Again we separate the p olynomial expre ssion P (1) n − 1 and ǫ n to g e t a second ob ject, which will b e called Ω n : Ω n = W ( P n ∂ x P (1) n − 2 − P (1) n − 1 ∂ x P n − 1 ) − V P n − 1 P (1) n − 1 − U P n P n − 1 (2.11a) = W ( ǫ n ∂ x P n − 1 − P n ∂ x ǫ n − 1 ) + V ǫ n P n − 1 (2.11b) W e can expr ess b oth Ω n and Θ n in terms o f the recurrence co efficients by s ubstituting the expres - sions for P n (2.5a) and ǫ n (2.5b) in to Ω n (2.11b) and Θ n (2.8b). Θ n = W ( x ) h n (" 1 x n +1 + n X j =0 S j ! 1 x n +2 + · · · # × " nx n − 1 − n − 1 X j =0 S j ! ( n − 1) x n − 2 + · · · # + " n + 1 x n +2 + n X j =0 S j ! n + 2 x n +3 + · · · # × " x n − n − 1 X j =0 S j ! x n − 1 + · · · #) + V ( x ) × h n " 1 x n +1 + n X j =0 S j ! 1 x n +2 + · · · # × " x n − n − 1 X j =0 S j ! x n − 1 + · · · # (2.12a) Ω n = W ( x ) ( h n " 1 x n +1 + n X j =0 S j ! 1 x n +2 + n X j =0 R j +1 + j X k =0 S j S k ! 1 x n +3 + · · · # × " ( n − 1) x n − 2 − ( n − 2) n − 2 X j =0 S j ! x n − 3 + ( n − 3) n − 2 X j =1 j − 1 X k =0 S j S k − R j ! x n − 4 + · · · # + h n − 1 " x n − n − 1 X j =0 S j ! x n − 1 + n − 1 X j =1 j − 1 X k =0 S j S k − R j ! x n − 2 + · · · # × " n x n +1 + n − 1 X j =0 S j ! ( n + 1) x n +2 + n − 1 X j =0 R j +1 + j X k =0 S j S k ! ( n + 2) x n +3 + · · · #) + V ( x ) × h n " 1 x n +1 + n X j =0 S j ! 1 x n +2 + n X j =0 R j +1 + j X k =0 S j S k ! 1 x n +3 + · · · # × " x n − 1 − n − 2 X j =0 S j ! x n − 2 + n − 2 X j =1 j − 1 X k =0 S j S k − R j ! x n − 3 + · · · # . (2.12b) These definitions will b e particular ly us eful when we ar e lo oking at examples of sp ecific semi- classical weigh ts. Since the recur r ence relatio n (1.2) can b e expresse d in a matrix for m ψ n +1 ( x ) =   x − S n − R n 1 0   ψ n ( x ) , where ψ n ( x ) =   P n ( x ) P n − 1 ( x )   (2.13) we collect the imp ortant r elations w e ha ve derived so far and put them in a matr ix form so that our int ended differential system ca n b e written as one e x pression. W e b egin with the t wo expressions 5 (2.8a) and (2.11a), written in matrix for m:   P n − 1 − P (1) n − 2 P n − P (1) n − 1     W ∂ x P (1) n − 1 W ∂ x P n   =   Ω n + V P n − 1 P (1) n − 1 + U P n P n − 1 Θ n + V P n P (1) n − 1 + U P 2 n   , (2.14a) which can easily b e solved making use of (2.4 a ) to giv e:   W ∂ x P (1) n − 1 W ∂ x P n   = 1 h n − 1   P (1) n − 1 − P (1) n − 2 P n − P n − 1     Ω n + V P n − 1 P (1) n − 1 + U P n P n − 1 Θ n + V P n P (1) n − 1 + U P 2 n   , (2.14b) so that w e have tw o differential eq uations: W ∂ x P n = 1 h n − 1 (Ω n P n − Θ n P n − 1 ) , (2.15a) W ∂ x P (1) n − 1 = (Ω n P (1) n − 1 − Θ n P (1) n − 2 + V h n − 1 P (1) n − 1 + U h n − 1 P n ) . (2.15b) Lo oking for a seco nd differ en tial rela tion for P n , w e take (2.15a) with a r educed index in conjunction with the recurr ence relation (1.2), which leads to W ( ∂ x P n − 1 ) = 1 h n − 2  Ω n − 1 P n − 1 − Θ n − 1 R n − 1 (( x − S n − 1 ) P n − 1 − P n )  . (2.16) How ever w e hav e no expr ession to remove the x from the equa tion, so we consider the pro blematic part of the expressio n: ( x − S n )Θ n = ( x − S n ) ( W ( ǫ n ∂ x ( P n ) − ∂ x ( ǫ n ) P n ) + V ǫ n P n ), which we expand using (1.2) and the differential o f (2.4b) to get: ( x − S n )Θ n = W ( − ∂ x ǫ n ( P n +1 + R n P n − 1 ) + ∂ x P n ( ǫ n +1 + R n ǫ n − 1 )) + V P n ( ǫ n +1 + R n ǫ n − 1 ) = Ω n +1 + R n Ω n + V h n (2.17) This allows us to remov e x from (2.16) to give a se c ond differential eq uation. W ∂ x P n − 1 = 1 h n − 1 (Θ n − 1 P n − Ω n P n − 1 ) − V P n − 1 (2.18) W e now hav e a differ ent ial system W ∂ x ψ ( x ) = 1 h n − 1   Ω n ( x ) − Θ n ( x ) Θ n − 1 ( x ) − (Ω n ( x ) + V ( x ) h n − 1 )   ψ ( x ) (2.19) where ψ ( x ) =   P n ( x ) P n − 1 ( x )   . Thus if w e g iv e the r ecurrence and differential equations in a se mi- discrete Lax repres en tation we hav e ψ n +1 ( x ) = L n ( x ) ψ n ( x ) (2.20a) ∂ x ψ n ( x ) = M n ( x ) ψ n ( x ) (2.20b) 6 where L n =   x − S n − R n 1 0   , M n = 1 W h n − 1   Ω n ( x ) − Θ n ( x ) Θ n − 1 ( x ) − (Ω n ( x ) + V ( x ) h n − 1 )   . Here we hav e identified the Lax ma trices L n and M n . So given a particular se mi-classical weight we can identif y the polyno mia ls V a nd W , which in turn lead to expressions for Θ and Ω. 2.2 Compatibilit y relations W e now use the differential sys tem (2.19) with the ma tr ix for m o f the recur rence relation (2.13) in order to create a compatibility relation so that relations b et ween Ω n and Θ n can b e derived. Thus we consider the c ompatibilit y b et ween the semi-dis crete Lax pa ir, which leads to the semi-discr ete Lax equation ∂ x L n = M n +1 L n − L n M n . (2.21) Equating this expressio n   1 0 0 0   = 1 W h n   Ω n +1 ( x ) − Θ n +1 ( x ) Θ n ( x ) − (Ω n +1 ( x ) + V ( x ) h n )     x − S n − R n 1 0   − 1 W h n − 1   x − S n − R n 1 0     Ω n ( x ) − Θ n ( x ) Θ n − 1 ( x ) − (Ω n ( x ) + V ( x ) h n − 1 )   (2.22) we can identif y tw o distinct re la tions ( x − S n )  Ω n +1 h n − Ω n h n − 1  = R n +1 Θ n +1 h n +1 − R n Θ n − 1 h n − 1 + W (2.23a) ( x − S n ) Θ n h n = Ω n +1 h n + Ω n h n − 1 + V , (2.23b) which we can identify a s b eing compara ble with the Lague r re-F reud e q uations [1]. Remark 2. 1 W e s ho uld p oin t o ut that this system could b e explo red independent o f orthogo nal po lynomials b y simply setting V = v 0 + v 1 x + v 2 x 2 + . . . + v n x n and W = w 0 + w 1 x + w 2 x 2 + . . . + w n x n (where the v j , w j are constants) and then see what differe nce equa tions are pro duced for different orders of V a nd W . How ever s ince we a re interested with the connections with semi-classica l orthogo nal p olynomials , we will pr esen t a semi-classic a l weigh t and then determine V and W . 7 3 A coupled differe nce equation and corresp onding third order nonlinear equation This metho d ca n b e demons trated by using a semi-clas s ical weigh t l 0 ( x ) = ( x − t ) α e − ( ax + b 2 x 2 ) synonymous with the (asso ciated) Lag uerre orthog onal p olynomials l ( x ) = x α e − x . Our choice of deformations for this weigh t, in volve a ltering the order of the p olynomial in the e xponential. W e first cons ider a deformation in the expo nen tial part of the w eight function, the semi-classica l weigh t w ( x ) = ( x − t ) α e − ( ax + b 2 x 2 ) with α, a, b > 0 and where the s upp ort S is an ar c fro m ( t → ∞ ). Then from the Pearson equatio n, we hav e V ( x ) = α − ( a + bx )( x − t ) , W ( x ) = x − t. (3.1) and from the consistency relatio ns we have tw o non-triv ial equations b ( R n +1 + R n ) = − S n [ bS n + ( a − bt )] + (2 n + 1 + at + α ) , (3.2a) R n +1 [ b ( S n +1 + S n ) + ( a − bt )] − R n [ b ( S n + S n − 1 ) + ( a − bt )] = S n − t. (3.2b) W e consider this to b e a nonlinear system in terms of the r ecurrence co efficients R n and S n , which has the linear system (2.20) with the La x pair : L n =   x − S n − R n 1 0   , (3.3a) M n = 1 x − t   n − bR n ( bx + a + b ( S n − t )) R n − ( bx + a + b ( S n − 1 − t )) bx 2 + x ( a − bt ) + bR n − n − α − at   (3.3b) for the asso ciated semi-discrete Lax equation (2 .21). This system can b e ca lle d a discrete integrable system due to the existence of the corr espo nding linear problem, i.e., the Lax pair. In the strictest sense w e ca nnot call the equatio n Painlev ´ e since it is third order howev er there are other ex amples of third order nonlinear difference equatio ns that a re k no wn to be integrable [4] . W riting this sys tem in matrix for m   b b b ( S n +1 + S n ) + ( a − bt ) − [ b ( S n + S n − 1 ) + ( a − bt )]     R n +1 R n   =   − S n ( bS n + ( a − bt )) + (2 n + 1 + at + α ) S n − t   (3.4) 8 allows us to solv e the system in terms o f R n +1 and R n and hence we can find a third order difference equation in S n  ( S n + S n − 1 +  a b − t  )( S n ( S n +  a b − t  )) − 1 b (2 n + 1 + at + α )(( S n + S n − 1 ) +  a b − t  ) − ( S n − t ) o × n − 2  S n +1 + a b − t  − ( S n + 2 S n +1 + S n +2 ) o =  ( S n +2 + S n +1 +  a b − t  )( S n +1 ( S n +1 +  a b − t  )) − 1 b (2 n + 1 + at + α )(( S n +2 + S n +1 ) +  a b − t  ) − ( S n +1 − t ) o × n − 2  S n + a b − t  − ( S n − 1 + 2 S n + S n +1 ) o . (3.5) Alternatively b y letting S n = Q n − Q n − 1 in (3.2), we are led to an alter nativ e third o rder difference equation in Q n that inc ludes an extra parameter c b  Q n − ( n + 1) t + c a − bt + b ( Q n +1 − Q n − 1 ) + Q n − 1 − nt + c a − bt + b ( Q n − Q n − 2 )  = − ( Q n − Q n − 1 )[ b ( Q n − Q n − 1 ) + a − bt ] + (2 n + 1 + at + α ) , (3.6) (where c is the constant of integration). 4 Conclusion and Outlo ok Given a class o f s emi-classical ortho g onal p olynomials (Hermite, Laguer re and Jaco bi), we can ident ify a semi-discrete Lax pair and th us an as socia ted dis crete integrable system. Using the Laguerr e weigh t l 0 ( x ) = ( x − t ) α e − ( ax + b 2 x 2 ) , we found a new coupled discrete in tegr able s ystem, which is first order in R and seco nd o rder in S . Com bining the tw o equa tions gives a thir d order difference equation in S or a third or de r difference equation in the new v ariable Q . Since we w ere only interested in the connections with disc r ete Painlev´ e equations, we hav e omitted to lo ok at the additional t -differ e n tial equation (whic h app ears as a consequence of the t pa r ameter in the weigh t function). It is likely that we could use the t -differ en tial equation to find a contin uous Painlev´ e equation related to our semi-cla s sical Lag ue r re weigh t. While we hav e c hosen to lo ok at a simple deforma tio n of the class ic al o rthogonal poly nomial weigh t asso ciated Laguerr e, this scheme can of course be used for der iving a multit ude o f disc r ete int egr able systems, through c ho osing an appropria te classical weigh t function. While w e ha ve made some pro gress in this regar d, finding the cor respo nding con tinuous equation is not a lw ays poss ible. W orking with further exa mples has shown that when lo oking for contin uum limits, the choice o f a semi-classica l weight function must b e of a particula r form. In this pap er we have applied the Lag uerre metho d to a family of class ical orthogo nal po ly- nomials, but we exp ect it is p ossible that the Laguerre metho d can b e used with other clas s es 9 of or thogonal p olynomials, such as the discr ete, multiple or q-orthog onal p olynomials. W e w ould need to alter the metho d appropr iately , such as choosing an analogue for the Pearson equation (since w e are using the “class ical Pearson eq uation” for cla s sical or thogonal p olynomials). Thu s, we could derive a similar sch eme for q-o rthogonal polynomia ls g iv en the q-Pearson e q uation, where a natural extension of this would b e to consider the q-La guerre or tho gonal p olynomials. 5 Ac kno wledgemen ts Paul Spicer wishes to thank Nalini Joshi for n umero us discussions on int egr able lattice equa tions, and for adv ice and encourag emen t. He wishes to thank Pa vlos Ka ssotakis for pro of r e ading the pap er and useful insights. 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