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Difference equations-congresses --- Symmetry(Physics)-Congresses --- Hypergeometric functions-congresses --- Difference equations --- Hypergeometric functions --- Symmetry (Physics)

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This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.

Differential equations --- Symmetry (Mathematics) --- 517.91 Differential equations --- Invariance (Mathematics) --- Physics. --- Algebra. --- Field theory (Physics). --- Difference equations. --- Functional equations. --- Numerical and Computational Physics, Simulation. --- Difference and Functional Equations. --- Field Theory and Polynomials. --- Group theory --- Automorphisms --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Equations, Functional --- Functional analysis --- Mathematics --- Mathematical analysis --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Field theory (Physics)

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Mathematical physics --- Painlevé equations --- Physique mathématique --- Asymptotic theory --- Congresses --- Théorie asymptotique --- Congrès --- Painleve equations --- 517.1 --- 517.58 --- Physical mathematics --- Physics --- Equations, Painlevé --- Functions, Painlevé --- Painlevé functions --- Painlevé transcendents --- Transcendents, Painlevé --- Differential equations, Nonlinear --- Introduction to analysis --- Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials. --- Mathematics --- 517.58 Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials. --- 517.1 Introduction to analysis --- Painlevé equations --- Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials --- Painleve equations - Congresses. --- Mathematical physics - Asymptotic theory - Congresses.