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Zbl 1082.42016
Ismail, Mourad E. H.
(Van Assche, Walter)
Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. (English)
[B] Encyclopedia of Mathematics and its Applications 98. Cambridge: Cambridge University Press. xviii, 706~p. \sterling~80.00; \$~140.00 (2005). ISBN 0-521-78201-5

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 Szeg\H{o}'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); ``\dots 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.
[Leonid Golinskii (Kharkov)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
42C15 Series and expansions in general function systems
33D80 Connections with groups, algebras and related topics
33C45 Orthogonal polynomials and functions of hypergeometric type
33C50 Hypergeometric functions and integrals in several variables
33D45 Basic hypergeometric functions and integrals in several variables
42-02 Research monographs (Fourier analysis)
33-02 Research monographs (special functions)

Keywords: classical orthogonal polynomials; discrete orthogonal polynomials; zeros and inequalities; polynomials on the unit circle; linearization and integral representation; $q$-orthogonal polynomials; Riemann-Hilbert problem; multiple orthogonal polynomials

Citations: Zbl 1082.42020; Zbl 1082.42021

Cited in: Zbl 1246.42022 Zbl 1223.42020 Zbl 1250.42082 Zbl 1172.42008

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