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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. (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 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.

MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 42C15 General harmonic expansions, frames 33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, $$p$$-adic groups, Hecke algebras, and related topics 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces 33-02 Research exposition (monographs, survey articles) pertaining to special functions