**CRM Proceedings & Lecture Notes**

Volume: 9;
1996;
388 pp;
Softcover

MSC: Primary 39;
Secondary 33

Print ISBN: 978-0-8218-0601-2

Product Code: CRMP/9

List Price: $119.00

AMS Member Price: $95.20

MAA member Price: $107.10

**Electronic ISBN: 978-1-4704-3923-1
Product Code: CRMP/9.E**

List Price: $119.00

AMS Member Price: $95.20

MAA member Price: $107.10

# Symmetries and Integrability of Difference Equations

Share this page *Edited by *
*Decio Levi; Luc Vinet; Pavel Winternitz*

A co-publication of the AMS and Centre de Recherches Mathématiques

This book is devoted to a topic that has undergone rapid
and fruitful development over the last few years: symmetries
and integrability of difference equations and
\(q\)-difference equations and the theory of special functions
that occur as solutions of such equations. Techniques that have been
traditionally applied to solve linear and nonlinear differential
equations are now being successfully adapted and applied to discrete
equations.

This volume is based on contributions made by leading experts in the
field during the workshop on Symmetries and Integrability of Difference
Equations held in Estérel, Québec, in May 1994.

Giving an up-to-date review of the current status of the field,
the book treats these specific topics: Lie group and quantum
group symmetries of difference and \(q\)-difference
equations, integrable and nonintegrable discretizations of continuous
integrable systems, integrability of difference equations,
discrete Painlevé property and singularity confinement,
integrable mappings, applications in statistical mechanics and field
theories, Yang-Baxter equations, \(q\)-special functions and
discrete polynomials, and \(q\)-difference integrable
systems.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students, research mathematicians and physicists working in difference equations, special function theory, applications of Lie groups theory, nonlinear phenomena in general and integrability in particular. Also of interest to pure and applied mathematicians, theoretical and mathematical physicists, and engineers interested in solitons.

# Table of Contents

## Symmetries and Integrability of Difference Equations

- Cover Cover11
- Title page v6
- Contents vii8
- Preface xi12
- On the numerics of integrable discretizations 114
- A brief introduction to the world of 𝑞 1326
- A Ramanujan-type for the Al-Salam and Ismail biorthogonal rational functions 2134
- Knizhnik-Zamolodchikov equations and the algebraic Bethe Ansatz 2942
- Integrable quantum mappings 3750
- Local master symmetries of differential-difference equations 5164
- Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations 6376
- On the diagonalization of difference Calogero-Sutherland systems 7992
- The integrable dynamics of a discrete curve 91104
- Continuous symmetries of finite-difference evolution equations and grids 103116
- Basic Bessel functions and 𝑞-difference equations 113126
- Difference equations and gauge symmetry 125138
- An operator-valued Riemann-Hilbert problem associated with the XXX model 133146
- Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation 143156
- Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations 155168
- Lie symmetries and linearizations of analytic discrete dynamical systems 163176
- 𝑞-algebra representations of the Euclidean, pseudo-Euclidean, and oscillator algebras, and their tensor products 173186
- ₃𝐹₂(1) hypergeometric function and quadratic 𝑅-matrix algebra 185198
- Lie point symmetries of differential difference equations 199212
- The factorization method for the differential-difference relativistic Schrödinger equation and 𝑞-deformations 209222
- Quantum deformations of 𝜏-functions, bilinear identities, and representation theory 219232
- The factorization method and hierarchies of 𝑞-oscillator Hamiltonians 239252
- Discrete-time Calogero-Moser model and lattice KP equations 253266
- Bilinear structure and exact solutions of the discrete Painlevé I equation 265278
- Integrable difference equations and numerical analysis algorithms 269282
- An integral representation of the very-well-poised _{8𝜓8} 281294
- Singular analogue of the Fourier transformation for the Askey-Wilson polynomials 289302
- Singularity confinement 303316
- Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials 319332
- Linearisation of difference equations using factorisable Lie symmetries 337350
- First integrals of the infinite Toda lattice 345358
- Non semisimple quantum groups and “exponential mappings” 353366
- Discrete Schrödinger equation, Darboux transformations, and orthogonal polynomials 363376
- Integrable cellular automata and AKNS hierarchy 371384
- On some rational Coxeter groups 377390
- Back Cover Back Cover1402