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.