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Basic Hypergeometric Series

George Gasper and Mizan Rahman
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and Its Applications 96
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Warren Johnson
, on

I was in graduate school when the first edition of this book appeared in 1990. I knew some of the subject matter from Dick Askey's special functions courses and seminars, and I had seen a preliminary version of the book at a conference at Ohio State in 1989, and I remember very well how excited I was when my copy arrived. I want to write "It did not disappoint," but actually that's not quite true; I was a dumb kid, and I was hoping it would have more on the combinatorial aspects of q-series than it does. Nevertheless I immediately set about reading it and doing problems, and I believe I was one of the first to send in a list of errata. I still have George Gasper's reply of 27 August 1990, thanking me for my list and commiserating with me on my struggles with exercise 5.4.

The book really was a major event in one part of mathematics, and it was very well received. George Andrews (American Mathematical Monthly, vol. 98, March 1991, pp. 282-284) wrote "Reviewers for the MONTHLY are told to write 'a chatty essay [that] ... does not have to be closely tied to the book under review.' At the risk of violating this quite sensible suggestion, I shall begin by saying, 'I love this book! It is great!'" Jet Wimp's review (SIAM Review, vol. 33, September 1991, pp. 489-493) was no less complimentary. More recently, Steven Milne has emphasized the importance of the book in both his research and his teaching (on p. 28 of Theory and Applications of Special Functions, a collection of papers dedicated to Mizan Rahman that I have also reviewed here).

Andrews and Wimp both stressed the pedagogical qualities of the book, particularly the large collection of exercises, and both noted that this was what set it apart from other books on q-series that appeared in the 1980s. This was especially charitable of Andrews in that these include q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, his own beautiful series of lectures, and Nathan Fine's equally beautiful Basic Hypergeometric Series and Applications, for which Andrews wrote the Foreword and Chapter Notes. Gasper and Rahman tried to write a useful book, rather than a beautiful one, and, as Andrews said, "they have masterfully pulled it off." We need both kinds.

The reviews of Andrews and Wimp are both excellent and readily available on JSTOR, so I will focus on what is new in this edition. Let me attempt a brief description of the subject first. The word "Basic", though standard, is apt to cause confusion. 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). In the most interesting cases there are as many shifted factorials in the numerator as in the denominator, except that the denominator always has an n!, so one typically speaks of 2-F-1's or 3-F-2's or 7-F-6's, with one more free parameter in the numerator. A 1-F-0 is the binomial series (1-x)-a.

One could define q-hypergeometric series by starting with q-numbers: let the q-analogue of the number a be [a] = (1-qa)/(1-q), where this means a if q = 1. Then a q-analogue of the shifted factorial would be something like [a][a+1]...[a+n-1]. Since there would typically be as many of these in the numerator as in the denominator, there are some factors of (1-q)n that could be cancelled, so one might as well look at quotients of products of the form


instead; or, better yet, products of the form

(a;q)n := (1-a)(1-aq)...(1-aqn-1),

which are more general (in particular, a can be zero), easier to print, and no more difficult to work with. This then is a basic (or q-) hypergeometric series — a power series where the coefficient c(n) of xn is a quotient of products of these q-shifted factorials (a;q)n; q is the "base", and typically (for convergence) |q| < 1. The two most interesting cases are (i) with (q;q)n in the denominator (analogous to the n! in ordinary hypergeometric series), and otherwise with one more more q-shifted factorial in the numerator than in the denominator; and (ii) without (q;q)n in the denominator, with as many q-shifted factorials in the denominator as in the numerator, and bilateral — summed from minus infinity to infinity. Many q-hypergeometric series identities reduce to ordinary hypergeometric series identities as q tends to 1, if one first rescales by replacing a's by q^a's, but many other q-series identities don't reduce to anything interesting as q tends to 1, so that they have a life all their own.


One could imagine another generalization, in which a is replaced not by [a] = (1-qa)/(1-q) but by := sin(as)/sin(s), which reduces to a as s tends to zero. (In practice s is taken in the form πσ.) Thus a trigonometric analogue of the shifted factorial would be something like ..., and a trigonometric hypergeometric series would be a power series whose coefficients are quotients of products of these trigonometric shifted factorials. This is, however, more or less equivalent to using products like [a][a+1]...[a+n-1], by taking q = e2is. So this idea, while intriguing at first sight, seems on reflection not to be very interesting. It really is interesting, though, because it hints at a further generalization.

There are three new chapters (9-11) in this edition, of which the last is by far the most interesting. It is about so-called elliptic hypergeometric functions, power series whose coefficients are quotients of products of shifted factorials built up as above from "elliptic numbers", which are themselves quotients of theta functions, and therefore (far-reaching) generalizations of the trigonometric numbers . (Alternatively, one could build these "elliptic" shifted factorials directly out of theta functions. A theta function is essentially a product of the form (x;q)(q/x;q), and one of the central results in q-series is Jacobi's triple product identity that expresses these infinite products as infinite series.)

These elliptic hypergeometric functions arose for the first time in the mid-1990s, in work by Frenkel and Turaev on a model in statistical mechanics, and this was rather like the discovery of a new Pacific island. The preface to the second edition lists 28 people who have done important work on these functions in the last 10 years. This is the most exciting development in q-series in that time, and the authors say that "a new edition could be justified only if we included a chapter" on them. Some new material (e.g., the q-numbers and the trigonometric numbers) has been added to chapter 1 as background for chapter 11.

Chapters 2-8 are largely unchanged. Several sections of chapter 2 are slightly improved by using the idea of a very-well-poised-balanced series. There are about 4 new problems in each chapter (more in chapter 8, less in chapter 4), added at the end so that all the original problems still have the same numbers. One of the strongest points of the book is the references, which have been updated (they now occupy pages 367-414), as have the bibliographical notes at the end of each chapter. To my mind, this also amply justifies a new edition.

The result of exercise 1.16 is often called Lebesgue's identity, though it is not accredited in this book. (This is V. A. Lebesgue, in a paper in Liouville's Journal in 1840, not Henri Lebesgue, then aged -35.) It appears without proof in Jacobi's epoch-making Fundamenta Nova of 1829, formula (8) in the 66th and final section. As Lebesgue and Jacobi both point out, it generalizes an identity of Gauss from 1808. Gauss's identity is also a special case of Jacobi's triple product identity, the subject of section 64 of the Fundamenta Nova.

The fact that the q-binomial coefficient generates partitions into at most k parts each at most n-k is often attributed to Sylvester's great paper on partitions of 1882, as it is here in the notes for exercise 1.2. It is actually due to Cayley, in his paper Researches on the Partition of Numbers in 1855. I did not try as hard to find misprints this time as I did 15 years ago, but there is an obvious one in formula (11.2.44), where all the q's should be p's.

This is an even better book than the first edition for several reasons: the up-to-date bibliography, the new material, and the opportunity to fix all the known errata. There are a few things one might wish for that aren't there, for example the connections to quantum groups, or the theory of hypergeometric series on the Lie group U(n) (due largely to Milne), or more combinatorics. But we can really only judge authors by what they do write, not what they don't. Gasper and Rahman have, once again, done admirably.

Warren Johnson ( is visiting assistant professor of mathematics at Connecticut College.

Foreword; Preface; 1. Basic hypergeometric series; 2. Summation, transformation, and expansion formulas; 3. Additional summation, transformation, and expansion formulas; 4. Basic contour integrals; 5. Bilateral basic hypergeometric series; 6. The Askey-Wilson q-beta integral and some associated formulas; 7. Applications to orthogonal polynomials; 8. Further applications; 9. Linear and bilinear generating functions for basic orthogonal polynomials; 10. q-series in two or more variables; 11. Elliptic, modular, and theta hypergeometric series; Appendices; References; Author index; Subject index; Symbol index.