Symmetries and Integrability of Difference Equations: (CRM Proceedings & Lecture Notes)

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 Esterel, Quebec, 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 Painleve 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.

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