Theory of Hypergeometric Functions

On page 231 of Indiscrete Thoughts, Gian-Carlo Rota’s fascinating collection of essays, one finds the following (to borrow one of Rota’s phrases) Sybilline pronouncement, which must originally have appeared in one of his book reviews in Advances in Mathematics:

Hypergeometric functions are one of the paradises of nineteenth-century mathematics that remain unknown to mathematicians of our day. Hypergeometric functions of several variables are an even better paradise: they will soon crop up in just about everything.

If events have not quite yet justified this prediction, it is largely because there are too many possible definitions. It is necessary to make progress in other areas of mathematics to see which multivariate objects that might plausibly claim to be hypergeometric are cropping up at least in something.

The book under review deals with this issue, and was originally published in Japan in 1994. The second author died — much too young — the following year. This translation, due to Kenji Iohara, appeared in 2010. While it won’t win any awards for English prose style, the sense is always clear. According to the first author “the contents are almost the same as the original except for a minor revision. (…) As for the references, we just added several that are directly related to the contents of this book.”

Aomoto is perhaps best known for his extension of the Selberg integral. I met him in graduate school one summer when he came to Wisconsin to visit Dick Askey; he lectured on this subject to an audience of two, which might tend to contradict the previous sentence, but anyway that’s how I will always think of him. His beautiful argument may be found in section 8.2 of Special Functions, by Andrews, Askey, and Roy.

The first chapter is a brief review of the univariate theory. In some introductory remarks on page 1 the authors observe that, although the Gamma function is not uniquely determined by the functional equation Γ(z+1)=zΓ(z), it can be found from this equation if we specify its behavior at infinity, and they add that “this is our basic idea to treat hypergeometric functions in this book.”

In section 1.2 the authors give the standard definition of a ₂F₁ hypergeometric series, but with some nonstandard notation. Hypergeometric series are built up from shifted factorials, which are products of the form a(a+1)…(a+n–1) for some complex number a, where this means 1 if n=0. Such products are usually abbreviated by (a)_n. Although these are often called Pochhammer symbols, this is not a very good name — Pochhammer used this notation, but for binomial coefficients, not for these products. The authors make the still more curious decision to use (a;n) instead of (a)_n and to nevertheless call this a Pochhammer symbol. (They also consistently use √–1 instead of i.) If one has the gamma function then one doesn’t actually need a new notation, since (a)_n = Γ(a+n)/Γ(a). In section 1.1 the authors make this quotient of gamma functions match Stirling’s formula, and thereby derive Euler’s infinite product for the Gamma function; this is an implementation of the idea of the previous paragraph.

A _pF_q hypergeometric series is a power series where the coefficient of xⁿ/n! has p shifted factorials in the numerator and q in the denominator. Except for the ₁F₀, which is the binomial series, the most common type is the ₂F₁, so much so that this book (and many other books, mostly older ones) suppresses the subscripts 2 and 1 after their first appearance. The gamma function is ubiquitous in the theory, e.g. in Euler’s integral representation (section 1.3), which gives the ₂F₁ as a combination of gamma functions times a generalization of the beta integral. The authors write “one of our purposes is to extend this integral to higher-dimensional cases and to reveal systematically the structure of generalized hypergeometric functions.”

At the end of this introductory chapter (page 19) the authors give a nice road map for the rest of the book. To carry out their program they require a superstructure of abstractions that one does not normally associate with special functions, and this is outlined in chapter 2: affine varieties, ordinary and twisted de Rham homology and cohomology, exact sequences of sheaves, logarithmic differential forms, filtrations, and arrangements of hyperplanes. One had better have a good background in most of the above (I regret to say that I do not) to expect to get much out of this book.

In chapter 3 the authors discuss a family of generalized hypergeometric series of type (n+1, m+1), where m and n are positive integers with n2F₁. More encouragingly, when n=1 and m=4 it reduces to a classical two variable generalization of the ₂F₁ due to Appell, and for n=1 and general m to a multivariate version of Appell’s series introduced by Lauricella. Some other bivariate hypergeometric series studied by Horn are also special cases. Integrals like that of Selberg arise as integral representations of these generalized hypergeometric functions over hyperplane arrangements.

In the fourth and final chapter the authors prove a classical theorem of G. D. Birkhoff on the asymptotic behavior of solutions of difference equations, and apply it to the foregoing theory. This is the idea foreshadowed on page 1.

The book also has four appendices. The first sketches the theory of a class of hypergeometric series slightly more general than the (n+1, m+1) type. The second contains a brief discussion of the Selberg integral. The third is background for the fourth, which was written by Toshitake Kohno. It brings in the Knizhnik–Zamolodchikov (KZ) equation and some other ideas from mathematical physics.

This book is an outstanding achievement, but it is not for the general reader, nor even for experts on classical hypergeometric series unless their mathematical education has been broader than mine.

Warren Johnson (warren.johnson@conncoll.edu) is visiting assistant professor of mathematics at Connecticut College.