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Theory and Applications of Special Functions

Mourad E. H. Ismail and Erik Koelink, editors
Springer Verlag
Publication Date: 
Number of Pages: 
Developments in Mathematics 13
[Reviewed by
Warren Johnson
, on
This is a collection of papers dedicated to Mizan Rahman, who is best known in the mathematical community for co-authoring with George Gasper the modern classic Basic Hypergeometric Series, a second edition of which has recently appeared. In this context "basic" doesn't mean what you might think. A hypergeometric series is a power series where, if the coefficient of xn is c(n), then c(n+1)/c(n) is a rational function of n. This implies that c(n) is a quotient of products of shifted factorials, which are themselves products of the form a(a+1)...(a+n-1). Basic hypergeometric series use products of the form (1-a)(1-aq)...(1-aqn-1) instead, where q is a complex number, so that c(n+1)/c(n) is now a rational function of qn; q is called the base, and typically |q|<1. Many q-series identities reduce to ordinary series identities as q tends to 1, though one often has to do a rescaling first. With hypergeometric series in particular, the theory carries over remarkably well to the q realm.

There are 21 research papers in the collection, some of which were presented at a special session at the AMS meeting in Baltimore in January 2003. Two of the papers are by Rahman, both co-authored with Gasper. There is also a valuable survey of Rahman's work by the editors and Richard Askey, and some other biographical material. I did not know that Rahman is an award-winning Bengali writer as well, and I think that a translation of one of his shorter essays would have been a nice addition. Rahman started out working in mathematical physics before migrating into special functions, and once he started thinking about q-series he never stopped. He soon became one of the world's foremost experts, a distinction that, thanks to Basic Hypergeometric Series, he will hold in perpetuity.

Thus it is fitting that most of the papers in the collection are on some aspect of q-series. Naturally I have my favorites: I would read George Andrews or Bruce Berndt if they wrote about sheep farming, there are several other authors here whose work I always enjoy, and I am especially glad to have the paper of Michael Schlosser. It would be presumptuous of me to make further judgments about the papers; suffice it to say that many world leaders in special functions have contributed, and the overall quality is very high. Anyone working in q-series should own this collection. It may also be of interest to people in other areas within or related to special functions.

Warren Johnson ( is visiting assistant professor of mathematics at Connecticut College. One of his favorite areas of mathematics is special functions.

Preface.- Dedication.- Mizan Rahman, His Mathematics and Literary Writings.- q-Bessel Functions.- a-Gaussian Polynomials.- On a Generalized Gamma Convolution Related to the q-Calculus.- Ramanujan and Cranks.- The Saalschtz Chain Reactions and Multiple q-Series Transformations.- Painlev‚ Equations and Associated Polynomials.- Zeta Functions of Heisenberg Graphs over Finite Rings.- q-Analogues of Some Multivariable Biorthogonal Polynomials.- Multivariable Askey-Wilson Polynomials.- Continuous Hahn Functions.- Series and Integrals.- Little q-Jacobi Functions.- A Second Addition Formula for Continuous q-Ultraspherical Polynomials.- A Bilateral Series Involving Basic Hypergeometric Functions.- Hilbert Space Asymptotics for Orthogonal Polynomials.- Generalizations of Jacobi's Triple Product Identity.- Summable Sums.- Askey-Wilson Functions and Quantum Groups.- Expansions in q-Polynomials.- Strong Nonnegative Linearization of Orthogonal Polynomials.- Remarks on Some Basic Hypergeometric Series.