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Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. (English) Zbl 1082.42016
Encyclopedia of Mathematics and its Applications 98. Cambridge: Cambridge University Press (ISBN 0-521-78201-5). xviii, 706 p. £ 80.00; $ 140.00 (2005).
The monograph by Mourad Ismail will meet the needs for an authoritative, up-to-date, self-contained and comprehensive account of the theory of orthogonal polynomials treated for the first time from the viewpoint of special functions. The coverage is encyclopedic, including classical topics originated in a celebrated Gabor Szegő’s book, as well as those, e.g. Askey-Wilson polynomial systems and $q$-orthogonal polynomials, discovered over the last several decades. Such topics as multiple orthogonal polynomials and the Riemann-Hilbert method applied to the asymptotics of such polynomials are exhibited for the first time in book form. Two chapters on orthogonal polynomials on the unit circle supplement nicely the recent monograph by {\it B. Simon} [“Orthogonal polynomials on the unit article. Part 1: Classical theory” (2005; Zbl 1082.42020); “… Part 2: Spectral theory” (2005; Zbl 1082.42021)]. A number of applications of the subject are presented including birth and death processes, integrable systems, combinatorics, and physical models. A chapter on open research problems and conjectures will definitely stimulate further research on the subject. An exhaustive bibliography rounds off the work. Simplicity, clarity of exposition, thoughtfully designed exercises are among book’s strengths. The book will be of interest not only to mathematicians, but also to a wide range of scientists and engineers.

42C05General theory of orthogonal functions and polynomials
42C15General harmonic expansions, frames
33D80Connections of basic hypergeometric functions with groups, algebras and related topics
33C45Orthogonal polynomials and functions of hypergeometric type
33C50Hypergeometric functions and integrals in several variables
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
42-02Research monographs (Fourier analysis)
33-02Research monographs (special functions)