Selected Title s i n Thi s Serie s

Volume 9 D . Levi, L . Vinet, an d P. Winternitz, Editor s Symmetries an d integrabilit y o f differenc e equation s 1996

8 J . Feldman , R . Froese , and L . M. Rosen , Editor s Mathematical quantu m theor y II : Schrodinger operator s 1995

7 J . Feldman , R . Froese , and L . M. Rosen, Editor s Mathematical quantu m theor y I : Field theor y an d many-bod y theor y 1994

6 Guid o Mislin, Edito r The Hilto n Symposiu m 199 3 Topics i n topology an d grou p theor y 1994

5 D . A. Dawson, Edito r Measure-valued processes , stochastic partial differentia l equations , and interactin g systems 1994

4 Hersh y Kisilevsk y an d M. Ra m Murty, Editor s Elliptic curve s an d relate d topic s 1994

3 Rem i Vaillancour t an d Andrei L . Smirnov, Editor s Asymptotic method s i n mechanic s 1993

2 Phili p D. Loewe n Optimal contro l vi a nonsmooth analysi s 1993

1 M . Ram Murty, Edito r Theta function s 1993 This page intentionally left blank Symmetries an d Integrability o f Difference Equation s This page intentionally left blank https://doi.org/10.1090/crmp/009 Volume 9 CRM PROCEEDINGS & LECTURE NOTES

Centre de Recherches Mathematique s Universite de Montrea l

Symmetries an d Integrability o f Difference Equation s

Decio Lev i Luc Vine t Pavel Winternit z Editors

The Centr e d e Recherche s Mathematique s (CRM ) o f th e Universite d e Montrea l wa s create d i n 196 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M i s supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , and th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) o f Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique .

American Mathematical Society Providence, Rhode Island US A The productio n o f thi s volum e wa s supporte d i n par t b y th e Fond s pou r l a Formation d e Chercheur s e t l'Aid e a la Recherche (Fond s FCAR) an d th e Natura l Sciences and Engineerin g Researc h Counci l o f Canada (NSERC) .

1991 Mathematics Subject Classification. Primar y 39A10 ; Secondary 33Dxx .

Library o f Congres s Cataloging-in-Publicatio n Dat a Symmetries an d integrabilit y o f differenc e equation s / D . Levi , L . Vinet , P . Winternitz, editors , p. cm . — (CR M proceeding s & lecture notes , ISS N 1065-8580 ; v. 9 ) Papers fro m th e Worksho p o n Symmetrie s and Integrabilit y o f Differenc e Equations , hel d Ma y 22-29, 1994 , i n Esterel , Quebec . Includes bibliographica l references . ISBN 0-8218-0601- 7 (alk . paper ) 1. Differenc e equations—Congresses . 2 . Hypergeometric functions—Congresses . 3 . Symme - try (Physics)—Congresses . I . Levi , D . II . Vinet, Luc . III . Winternitz, Pavel . IV . Worksho p on Symmetrie s an d Integrabilit y o f Differenc e Equation s (199 4 : Esterel, Quebec ) V . Series . QA431.S95 199 6 515'.625—dc20 96-2679 6 CIP

Copying and reprinting. Materia l in this book may be reproduced b y any means fo r educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r delivery o f documents and provide d that th e customary acknowledgmen t o f the source i s given. Thi s consent doe s not extend to other kind s o f copying fo r general distribution, fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o [email protected]. Excluded fro m thes e provision s i s material i n article s fo r whic h th e autho r hold s copyright . I n such cases , requests fo r permissio n t o us e or reprint shoul d b e addressed directl y to the author(s) . (Copyright ownershi p i s indicated i n the notic e i n the lowe r right-han d corne r o f the first pag e o f each article. ) © 199 6 by th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This volum e wa s typeset usin g AMS-T^K, the America n Mathematica l Society' s T^i macr o system , and submitte d t o the America n Mathematica l Societ y i n camera-read y form b y the Centr e d e Recherche s Mathematiques .

10 9 8 7 6 5 4 3 2 1 0 1 00 9 9 9 8 9 7 9 6 Contents

Preface Decio Levi, Luc Vinet, and Pavel Winternitz vi i On th e Numeric s o f Integrabl e Discretization s M. J. Ablowitz, B. M. Herbst, and C. Schober 1 A Brie f Introductio n t o th e Worl d o f q Richard Askey 1 3 A Ramanujan-Type Measur e fo r the Al-Sala m an d Ismai l Biorthogonal Rationa l Function s Natig M. Atakishiyev 2 1 Knizhnik-Zamolodchikov Equation s an d th e Algebrai c Bethe Ansat z H. M. Babujian and R. Flume 2 9 Integrable Quantu m Mapping s H. W. Capel and F. W. Nijhoff 3 7 Local Maste r Symmetrie s o f Differential-Differenc e Equations Igor Cherdantsev and Ravil Yamilov 5 1 Backlund Transformation s an d Hierarchie s o f Exac t Solutions fo r th e Fourt h Painlev e Equatio n an d thei r Application t o Discret e Equation s Peter A. Clarkson and Andrew P. Bassom 6 3 On th e Diagonalizatio n o f Differenc e Calogero-Sutherland System s J. F. van Diejen 7 9 The Integrabl e Dynamic s o f a Discrete Curv e Adam Doliwa and Paolo Maria Santini 9 1 Continuous Symmetrie s o f Finite-Differenc e Evolutio n Equations an d Grid s Vladimir Dorodnitsyn 10 3 Basic Besse l Function s an d g-Differenc e Equation s Roberto Floreanini and Luc Vinet 11 3 Difference Equation s an d Gaug e Symmetr y D. V. Fursaev and V. G. Kadyshevsky 12 5 CONTENTS

An Operator-Value d Riemann-Hilber t Proble m Associated wit h th e XX X Mode l Holger Frahm, Alexander R. Its, and Vladimir E. Korepin 13 3 Orthogonal Polynomial s Satisfyin g Differentia l Equations: Th e Rol e o f the Darbou x Transformatio n F. Alberto Grilnbaum and Luc Haine 14 3 Quantum Isomonodromi c Deformation s an d th e Knizhnik-Zamolodchikov Equation s J. Hamad 15 5 Lie Symmetrie s an d Linearization s o f Analyti c Discret e Dynamical System s Nalini Joshi and Peter J. Vassiliou 16 3 q-Algebra Representation s o f the Euclidean , Pseudo-Euclidean an d Oscillato r Algebras , an d thei r Tensor Product s E. G. Kalnins and Willard Miller, Jr. 17 3 3l<2(l) Hypergeometri c Functio n an d Quadrati c .R-Matrix Algebr a Vadim B. Kuznetsov 18 5 Lie Poin t Symmetrie s o f Differentia l Differenc e Equations Decio Levi and Pavel Winternitz 19 9 The Factorizatio n Metho d fo r the Differential-Differenc e Relativistic Schrodinge r Equatio n an d ^-Deformation s R. M. Mir-Kasimov 20 9 Quantum Deformation s o f r-Functions , Bilinea r Identities an d Representatio n Theor y A. Mironov 21 9 The Factorization Method an d Hierarchies o f g-Oscillato r Hamiltonians Javier Negro 23 9 Discrete-Time Calogero-Mose r Mode l an d Lattic e K P Equations F. W. Nijhoff and G. D. Pang 25 3 Bilinear Structur e an d Exac t Solution s o f th e Discret e Painleve I Equatio n Yasuhiro Ohta, Kenji Kajiwara, and Junkichi Satsuma 26 5 Integrable Differenc e Equation s an d Numerica l Analysi s Algorithms V. Papageorgiou, B. Grammaticos, and A. Ramani 26 9 An Integra l Representatio n o f the Very-Well-Poise d g^ s Series Mizan Rahman 28 1 Singular Analogu e o f the Fourie r Transformatio n fo r th e Askey-Wilson Polynomial s Mizan Rahman and Sergei K. Suslov 28 9 CONTENTS

Singularity Confinemen t A. Ramani, B. Grammaticos, and V. Papageorgiou 30 3 Recurrence Relatio n Approac h fo r Connectio n Coefficients. Application s t o Classica l Discret e Orthogonal Polynomial s A. Ronveaux, S. Belmehdi, E. Godoy, and A. Zarzo 31 9 Linearisation o f Differenc e Equation s Usin g Factorisabl e Lie Symmetrie s R. Sahadevan, G. B. Byrnes, and G. R. W. Quispel 33 7 First Integral s o f the Infinit e Tod a Lattic e A. B. Shabat 34 5 Non Semisimpl e Quantu m Group s an d "Exponentia l Mappings" E. Sorace 35 3 Discrete Schrodinge r Equation , Darbou x Transformations, an d Orthogona l Polynomial s Vyacheslav Spiridonov, Luc Vinet, and Alexei Zhedanov 36 3 Integrable Cellula r Automat a an d AKN S Hierarch y Leon A. Takhtajan 37 1 On Som e Rationa l Coxete r Group s daude M. Viallet 37 7 This page intentionally left blank Preface

The "Worksho p o n Symmetrie s an d Integrabilit y o f Differenc e Equations" , or - ganized b y the Centr e d e recherche s mathematiques , Universit e d e Montreal , an d sponsored b y th e CRM , NSER C o f Canad a an d th e NSF , too k plac e i n Esterel , Quebec, Ma y 22-29 , 1994 . The purpos e wa s to ge t togethe r a grou p o f scientists activel y workin g i n th e field to revie w the presen t statu s an d discus s expected futur e development s i n this highly topical field. A secondary purpose was to produce these Workshop Proceed - ings tha t woul d no t jus t b e a collectio n o f individua l articles , bu t a monograp h reviewing a well-define d an d restricte d topi c fro m complementar y point s o f view. The worksho p attracte d 6 4 active participant s fro m 1 5 countries (ordere d ac - cording to the number o f participants: Canad a (18) , USA (15), Russia (7) , Italy (4), Netherlands (4) , Australia (3) , France (3) , Germany (3) , Azerbaijan, Belgium , Eng- land, Japan , Mexico , Spain , Ukrain e ( 1 each). The topic s treated fal l roughl y int o the followin g (overlapping ) categories : Lie group and quantum group symmetries of difference, q-difference and dif- ferential difference equations. G . Byrnes , V . A . Dorodnitsyn , R . Floreanin i an d L. Vinet, W . Hereman , N . Joshi, D . Lev i and P . Winternitz, J . Negro ; Chaos in integrable systems, integrable and non-integrable discretizations of integrable systems. M . Ablowitz , C . M . Schober ; Integrability of difference equations, discrete Painleve property, singularity con- finement, integrable mappings. H . W. Capel , P . A . Clarkson , F . Nijhoff , V . Papa - georgiou, A . Ramani, J . Satsuma , A . Shabat, R . Yamilov ; Applications in statistical mechanics and field theories, Yang-Baxter equations. R. Flume, P. G. O. Freund, J . Harnad, A . Its, V. G. Kadyshevsky, R . Mir-Kasimov , C. Viallet ; Cellular automata. P . M. Santini, L . Takhtajan ; q-difference equations and q-special functions, discrete polynomials, quantum groups. R . Askey , N. Atakishiyev, F . A . Grunbaum, H . T. Koelink , D . R. Masson , W. J . Miller , S . Moulazade , M . Rahman, A . Ronveaux , E . Sorace , V . Spiridonov , S. K. Suslov , A . Zhedanov ; q-difference integrable systems and other physical applications. J . F. van Diejen, V. Kuznetsov, A . Mironov . The workshop ended with a round table discussion, le d by Martin Kruskal, who also summed u p the presente d results . An importan t featur e o f th e worksho p wa s th e intensiv e interactio n betwee n mathematicians an d physicist s workin g i n differen t bu t correlate d fields. Indeed ,

xi xn PREFACE the backgrounds o f the participants included : Specia l function theory , i n particular basic hypergeometri c function s an d relate d orthogona l polynomials . Grou p theor y and algebra : Li e groups, discrete groups , quantum groups . Classica l an d quantu m integrable systems , finite an d infinite-dimensiona l ones , continuou s an d discret e ones. Elementar y particl e physics , statistical physics , classica l hydrodynamics an d many others . The meeting was considered a success by all participants. Th e field is a rapidl y developing one and it is to be hoped that this workshop will serve as the beginning of a serie s o f regular biannual meetings . A s a matter o f fact a second "Symmetr y an d Integrability o f Difference Equations " (SIDE ) meeting will be held at the Universit y of Kent a t Canterbur y i n 1996 .

Decio Lev i Luc Vine t Pavel Winternit z