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Symmetries and Integrability of Difference Equations
Edited by: Decio Levi, University of Rome III, Italy, and Luc Vinet and Pavel Winternitz, University of Montreal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Proceedings & Lecture Notes
1996; 388 pp; softcover
Volume: 9
ISBN-10: 0-8218-0601-7
ISBN-13: 978-0-8218-0601-2
List Price: US$119
Member Price: US$95.20
Order Code: CRMP/9
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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.


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

  • M. J. Ablowitz, B. M. Herbst, and C. M. Schober -- On the numerics of integrable discretizations
  • R. A. Askey -- A brief introduction to the world of \(q\)
  • N. M. Atakishiyev -- A Ramanujan-type for the Al-Salam and Ismail biorthogonal rational functions
  • H. M. Babujian and R. Flume -- Knizhnik-Zamolodchikov equations and the algebraic Bethe Ansatz
  • H. W. Capel and F. W. Nijhoff -- Integrable quantum mappings
  • I. Cherdantsev and R. Yamilov -- Local master symmetries of differential-difference equations
  • P. A. Clarkson and A. P. Bassom -- Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations
  • J. F. van Diejen -- On the diagonalization of difference Calogero-Sutherland systems
  • A. Doliwa and P. M. Santini -- The integrable dynamics of a discrete curve
  • V. Dorodnitsyn -- Continuous symmetries of finite-difference evolution equations and grids
  • R. Floreanini and L. Vinet -- Basic Bessel functions and \(q\)-difference equations
  • D. V. Fursaev and V. G. Kadyshevsky -- Difference equations and gauge symmetry
  • H. Frahm, A. R. Its, and V. E. Korepin -- An operator-valued Riemann-Hilbert problem associated with the XXX model
  • F. A. Grunbaum and L. Haine -- Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation
  • J. P. Harnad -- Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations
  • N. Joshi and P. J. Vassiliou -- Lie symmetries and linearizations of analytic discrete dynamical systems
  • E. G. Kalnins and W. Miller -- \(q\)-algebra representations of the Euclidean, pseudo-Euclidean, and oscillator algebras, and their tensor products
  • V. B. Kuznetsov -- \(_3F_2\)(1) hypergeometric function and quadratic \(R\)-matrix algebra
  • D. Levi and P. Winternitz -- Lie point symmetries of differential difference equations
  • R. M. Mir-Kasimov -- The factorization method for the differential-difference relativistic Schrödinger equation and \(q\)-deformations
  • A. D. Mironov -- Quantum deformations of \(\tau\)-functions, bilinear identities, and representation theory
  • J. Negro -- The factorization method and hierarchies of \(q\)-oscillator Hamiltonians
  • F. W. Nijhoff and G. D. Pang -- Discrete-time Calogero-Moser model and lattice KP equations
  • Y. Ohta, K. Kajiwara, and J. Satsuma -- Bilinear structure and exact solutions of the discrete Painlevé I equation
  • V. Papageorgiou, B. Grammaticos, and A. Ramani -- Integrable difference equations and numerical analysis algorithms
  • M. Rahman -- An integral representation of the very-well-poised \(_{8{\psi}8}\)
  • M. Rahman and S. K. Suslov -- Singular analogue of the Fourier transformation for the Askey-Wilson polynomials
  • A. Ramani, B. Grammaticos, and V. Papageorgiou -- Singularity confinement
  • A. Ronveaux, S. Belmehdi, E. Godoy, and A. Zarzo -- Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials
  • R. Sahadevan, G. B. Byrnes, and G. R. W. Quispel -- Linearisation of difference equations using factorisable Lie symmetries
  • A. B. Shabat -- First integrals of the infinite Toda lattice
  • E. Sorace -- Non semisimple quantum groups and ëxponential mappings\"
  • V. P. Spiridonov, L. Vinet, and A. S. Zhedanov -- Discrete Schrödinger equation, Darboux transformations, and orthogonal polynomials
  • L. A. Takhtajan -- Integrable cellular automata and AKNS hierarchy
  • C. M. Viallet -- On some rational Coxeter groups
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