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# Symmetries and Integrability of Difference Equations

### About this Title

**Decio Levi**, *University of Rome III, Rome, Italy*, **Luc Vinet**, *University of Montreal, Montreal, QC, Canada* and **Pavel Winternitz**, *University of Montreal, Montreal, QC, Canada*, Editors

Publication: CRM Proceedings and Lecture Notes

Publication Year:
1996; Volume 9

ISBNs: 978-0-8218-0601-2 (print); 978-1-4704-3923-1 (online)

DOI: https://doi.org/10.1090/crmp/009

MathSciNet review: MR1416820

MSC: Primary 39-06; Secondary 00B25, 33-06

### Table of Contents

**Front/Back Matter**

**Chapters**

- On the numerics of integrable discretizations
- A brief introduction to the world of $q$
- A Ramanujan-type for the Al-Salam and Ismail biorthogonal rational functions
- Knizhnik-Zamolodchikov equations and the algebraic Bethe Ansatz
- Integrable quantum mappings
- Local master symmetries of differential-difference equations
- Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations
- On the diagonalization of difference Calogero-Sutherland systems
- The integrable dynamics of a discrete curve
- Continuous symmetries of finite-difference evolution equations and grids
- Basic Bessel functions and $q$-difference equations
- Difference equations and gauge symmetry
- An operator-valued Riemann-Hilbert problem associated with the XXX model
- Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation
- Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations
- Lie symmetries and linearizations of analytic discrete dynamical systems
- $q$-algebra representations of the Euclidean, pseudo-Euclidean, and oscillator algebras, and their tensor products
- $_3F_2$(1) hypergeometric function and quadratic $R$-matrix algebra
- Lie point symmetries of differential difference equations
- The factorization method for the differential-difference relativistic Schrödinger equation and $q$-deformations
- Quantum deformations of $\tau $-functions, bilinear identities, and representation theory
- The factorization method and hierarchies of $q$-oscillator Hamiltonians
- Discrete-time Calogero-Moser model and lattice KP equations
- Bilinear structure and exact solutions of the discrete Painlevé I equation
- Integrable difference equations and numerical analysis algorithms
- An integral representation of the very-well-poised $_{8{\psi }8}$
- Singular analogue of the Fourier transformation for the Askey-Wilson polynomials
- Singularity confinement
- Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials
- Linearisation of difference equations using factorisable Lie symmetries
- First integrals of the infinite Toda lattice
- Non semisimple quantum groups and “exponential mappings”
- Discrete Schrödinger equation, Darboux transformations, and orthogonal polynomials
- Integrable cellular automata and AKNS hierarchy
- On some rational Coxeter groups