The first and third authors are well known for *The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue* (hereafter referred to as **KS**), which first appeared in 1994 as a technical report of the Delft University of Technology, and has been a standard reference for workers in orthogonal polynomials ever since. A recent version is on the arXiv. The book under review subsumes **KS** but is more ambitious. It seeks not only to give the most important facts for all the known sequences of hypergeometric and q-hypergeometric orthogonal polynomials, but also to show that there are no others.

The book starts with a brief but valuable foreword by Tom Koornwinder on the history of the classification problem for orthogonal polynomials. The word “classical” has a technical meaning in this subject, for which Koornwinder quotes a paper of George Andrews and Richard Askey: “A set of orthogonal polynomials is classical if it is a special case or a limiting case of the Askey-Wilson polynomials or the q-Racah polynomials.” The point of this definition is that a theorem of Douglas Leonard on association schemes implies that any hypergeometric or q-hypergeometric orthogonal polynomial sequence must be classical. (The q-Racah and Askey-Wilson polynomials are in some sense the same, both being terminating 4-phi-3 q-hypergeometric series with four free parameters besides the ubiquitous q, but with different normalizations and different choices of the variable.)

It is not difficult to see that any set of orthogonal polynomials {p_{n}(x)} must satisfy a three-term recurrence relation that expresses xp_{n}(x) as a linear combination of p_{n+1}(x), p_{n}(x) and p_{n–1}(x). Orthogonality implies that we don’t need p_{n–2}(x), …, p_{0}(x) in the combination, in the following way. We could find the coefficient of p_{j}(x) by integrating xp_{n}(x)p_{j}(x) against the orthogonality measure, which would mean doing a sum in the case of a discrete measure. But if p_{n}(x) is an orthogonal polynomial, then it is orthogonal to any polynomial of lower degree, because such a polynomial can be written as a linear combination of p_{0}(x), … , p_{n–1}(x). Since xp_{j}(x) is of lower degree than p_{n}(x) unless j≥n–1, the coefficient of p_{j}(x) must be zero for j≤n–2. If the orthogonality is with respect to a positive measure, as one normally assumes, a refinement of this argument gives a simple relationship between the signs of the coefficients of p_{n+1}(x) and p_{n–1}(x).

A deeper result is that the converse is true: given a three-term recurrence relation and this simple relationship, {p_{n}(x)} must be orthogonal with respect to some positive measure. This fact, usually called Favard’s theorem, makes it trivial to write down examples of orthogonal polynomial sequences, so the classification problem would be hopeless without some restriction like ``classical” or ``hypergeometric”. With such restriction, it amounts to asking which choices of parameters in the Askey-Wilson polynomials retain the positivity of the measure, and what limits can be taken. Some of these limits lead to the famous polynomials of Hermite, Laguerre, and Jacobi (and Legendre, a special case of Jacobi), which Koornwinder proposes to call the very classical orthogonal polynomials.

One would like to know not only what the polynomials are, but also what the corresponding measure is — a nontrivial problem, in general. A major project of the second author in a series of papers from 1986 to 2001 was to work out systematically what the possibilities are for the polynomials and the measures. Of course there is much earlier work along similar lines, but Lesky has added greatly to our knowledge of finite systems of orthogonal polynomials, which can arise as eigenfunctions of second order differential or difference operators.

The book under review integrates Lesky’s work with **KS**. This makes it a complete guide to the hypergeometric and q-hypergeometric orthogonal polynomials: it describes what is, and explains why what isn’t, isn’t. It consists of two Parts plus three preliminary chapters.

Chapter 1 contains the necessary background on special functions: the gamma and beta functions, some other integrals, hypergeometric series, and q-analogues of all of the above. I met an old friend in chapter 2, an operator of Wolfgang Hahn natural enough to have been rediscovered by yours truly in graduate school, and used in my thesis. (If you try to write down a common generalization of the q-derivative and the forward difference of stepsize h, you’ll rediscover it too.) This chapter finds the polynomial solutions of an eigenvalue problem associated with Hahn’s operator. Chapter 3 uses Favard’s theorem to show that these polynomials are orthogonal, and shows further that they satisfy a Rodrigues type formula, which means, very loosely, that the nth one is an nth derivative or nth difference of something.

Part I treats the Askey scheme. Chapter 4 finds all the orthogonal polynomials in the case where the eigenvalue problem of chapter 2 reduces to a second order differential equation. (This amounts to taking q=1 and h=0 in Hahn’s operator.) These are the very classical polynomials of Hermite, Laguerre and Jacobi, plus three finite systems discovered in 1929 by V. Romanovski. In chapters 5–8 the eigenvalue problem becomes a second order difference equation and we get polynomials of Charlier, Meixner, and Hahn, among others. Chapter 9 gives the essential properties of the polynomials found in chapters 4–8. It is essentially the same as chapters 1 and 2 of **KS**.

Part II treats the q-Askey scheme, so it is the q-extension of Part I, in which the eigenvalue problem of chapter 2 is a q-difference equation. Chapters 10–13 are concerned with identifying the polynomial solutions and the corresponding measures. Chapter 14, which is essentially the same as chapters 3–5 of **KS**, gives the essential properties of the polynomials and shows how they reduce to the ones in chapter 9 when q=1. This last is not as simple as it may sound, because it is usually necessary to rescale first. The building blocks of hypergeometric functions are expressions called shifted factorials, which have the form a(a+1)(a+2)…(a+n–1), whereas q-hypergeometric functions are built out of q-shifted factorials having the form

(1–a)(1–aq)…(1–aq^{n–1}).

The key fact in doing the conversion is that (1–q^{a})/(1–q) reduces to a when q tends to 1.

The book under review cannot really be considered a textbook on orthogonal polynomials, even hypergeometric ones, since it is so narrowly focused on the classification problem and completely devoid of applications. It would however be the ideal text for a graduate course devoted to the classification, and it is a valuable reference, which everyone who works in orthogonal polynomials will want to own.

Warren Johnson ([email protected]) is visiting assistant professor of mathematics at Connecticut College.