You are here

Classical and Quantum Orthogonal Polynomials in One Variable

Mourad E. H. Ismail
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and Its Applications 98
[Reviewed by
Warren Johnson
, on

The classic book on this subject is Gabor Szegö’s Orthogonal Polynomials, which came out in 1939. It was revised in 1958, but not very extensively (“Only a few particularly interesting new items have been added…”), and the most recent edition appeared in 1975. Quite a lot has happened in orthogonal polynomials since 1939, and even since 1975.

The classical orthogonal polynomials, or at least quantities closely related to them, occur as matrix elements in representations of some of the classical Lie groups. N. J. Vilenkin’s Special Functions and the Theory of Group Representations is still the best reference for this, although it too could stand updating, as there have been many further developments since it appeared in 1965, due largely to the Dutch school led by Tom Koornwinder. The word “quantum” in the title of the book under review is an allusion to this, in that q-analogues of the classical orthogonal polynomials occur in the representation theory of quantum groups.

We now know many more examples of orthogonal polynomials than Szegö did, but more is true. We can now claim to know all the hypergeometric and q-hypergeometric orthogonal polynomials of one variable — in other words, all the single variable orthogonal polynomials that have nice formulas. A complete classification came in the early 1980s from an unexpected source: a theorem on association schemes in the Ph.D thesis of Douglas Leonard at Ohio State University. This material has been elegantly reworked by Paul Terwilliger, and a brief account of Terwilliger’s theory is in section 20.4 of the present book. The most general orthogonal polynomials of this type are the Askey-Wilson polynomials, which were found around the same time as Leonard’s theorem. They have four free parameters in addition to the polynomial variable and the ubiquitous q, and all the other hypergeometric and q-hypergeometric orthogonal polynomials (in particular the classical polynomials of Jacobi, Legendre, Hermite and Laguerre) are special or limiting cases of them. Jacques Labelle created a beautiful poster depicting these transitions, and a complete guide to this so-called Askey scheme of hypergeometric and q-hypergeometric orthogonal polynomials was produced by Roelof Koekoek and René Swarttouw in 1994.

Richard Askey is also the connecting link between Gabor Szegö and the present author (and, for that matter, the reviewer). He edited Szegö’s Collected Papers, and wrote the Foreword of the book under review. Mourad Ismail did postdoctoral work with Askey in the 1970s.

Combinatorialists have had another role to play in our story besides that of taxonomist. A major project of the 1970s and 1980s was to give combinatorial interpretations and proofs of many identities for the classical orthogonal polynomials. This was done largely by French and French Canadian mathematicians such as Dominique Foata, Gérard Viennot, Andre Joyal, Jacques Labelle, François Bergeron, Gilbert Labelle and Pierre Leroux. The most comprehensive account is Combinatorial Species and Tree-like Structures (1998) by the latter three authors, and Viennot’s Une Théorie Combinatoire de Polynômes Orthogonaux Generaux (1983) has an alternative approach. The book under review gives only a few connections with combinatorics, which is however a few more than Szegö has.

Thus the theory of orthogonal polynomials has an algebraic side and a combinatorial side, in addition to the analysis and special functions sides that one finds in Szegö. I do not know whether there will ever be an author capable of giving a coherent and well-balanced account of all of these aspects. Perhaps Pólya and Szegö together could have done it if they had been born 50 years later; or Vilenkin might have with a Fountain of Youth and the right co-author.

Mourad Ismail is not Pólya/Szegö, but he is a fine mathematician, with an encyclopedic knowledge of the literature, whose own contributions to it are many and varied. He has given us the best book on general orthogonal polynomials since Szegö. At least on the special functions side, it is the best we are likely to have for some time. (For polynomials orthogonal on the unit circle, which get one chapter from Ismail and one from Szegö, the recent treatise by Barry Simon is the current standard.)

Ismail’s book began life as a set of notes for a graduate course on the Askey-Wilson polynomials at the University of South Florida. This would be roughly chapters 11-16 of the present book, with closely related material in chapters 17 and 18. The treatment is novel in that the requisite formulas for the summation of q-series are developed from the theory of the Askey-Wilson operator in chapter 12. While not the most elementary approach, this might be the most natural one if you are aiming at the material of the succeeding chapters. Chapter 13 discusses the q-Hermite and q-ultraspherical polynomials. Although the q-Hermite are the simplest q-orthogonal polynomials, the theory is already rich enough to furnish a proof of the celebrated Rogers-Ramanujan identities. Chapter 14 brings in q-exponential functions and related topics, and chapter 15 builds up to the Askey-Wilson polynomials. Chapter 16 contains further development of the Askey-Wilson operator with application to the eponymous polynomials.

The book has more than enough material for a year-long graduate course on orthogonal polynomials, although it could use more exercises. Chapters 1-9 correspond very roughly to Szegö’s book, and one would probably do some subset of them in the first semester before going on to the Askey-Wilson polynomials. In both books, chapter 1 has some miscellaneous results on analysis and chapter 2 gives basic results on orthogonal polynomials. The intersection is substantial, but there are also many differences. Both third chapters more or less continue chapter 2; here the intersection is practically empty, but some of the material on discriminants in Ismail’s chapter 3 is in chapter 6 of Szegö. Ismail follows Szegö in beginning chapter 4 with Jacobi polynomials, which are the most general of the classical orthogonal polynomials, and working downwards toward the ultraspherical, Legendre, Laguerre and Hermite polynomials. After this the books diverge more. Ismail also has some specialized topics in chapters 10 and 19-23, and chapter 24 gives some open problems. Chapters 22 and 23 were mostly written by Walter Van Assche.

Some years ago I attended a talk by Doron Zeilberger, with many experts on special functions in the audience. As usual Zeilberger was stimulating and entertaining, but not overprepared, and at one point he asked Ismail for a certain formula. Ismail supplied it without hesitation, and then, after a pause, asked “Why did you ask me, instead of … [he indicated several of the other experts]?” “Because I knew you’d get it right,” Zeilberger replied.

I am sorry to have to report that, despite the author’s well-deserved reputation for accuracy, his book is not free of misprints and other infelicities. Perhaps this is inevitable in an age where most mathematics books receive little or no editing. I am trying to write a book myself, and I know how hard it’s been for me to get all the bugs out of it. The best that can be said of the author’s prose style is that he is not writing in his first language, but I’ve seen a lot worse.

I’ll conclude with another anecdote, which I heard from Dick Askey. Some readers of this review will remember a program on Ramanujan that ran on PBS about 20 years ago. At one point George Andrews talks about how hard it is to discern Ramanujan’s thought processes from his work. In contrast, he says something to the effect that when one of his colleagues proves a theorem, he admires it, and he might even say “why didn’t I think of that?”, but he can usually see how the ideas arose in his colleague’s mind. Askey asked Andrews whether he had been thinking of a certain argument of Ismail when he said “why didn’t I think of that?”, and Andrews replied “of course”. It may be that only an expert in q-series would be so impressed, but for the interested reader it’s the proof of Theorem 12.3.1 on page 309.

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

 1. Preliminaries; 2. Orthogonal polynomials; 3. Differential equations; 4. Jacobi polynomials; 5. Some inverse problems; 6. Discrete orthogonal polynomials; 7. Zeros and inequalities; 8. Polynomials orthogonal on the unit circle; 9. Linearization, connections and integral representations; 10. The Sheffer classification; 11. q-series preliminaries; 12. q-summation theorems; 13. Some q-orthogonal polynomials; 14. Exponential and q-Bessel functions; 15. The Askey-Wilson polynomials; 16. The Askey-Wilson operators; 17. q-Hermite polynomials on the unit circle; 18. Discrete q-orthogonal polynomials; 19. Fractional and q-fractional calculus; 20. Polynomial solutions to functional equations; 21. Some indeterminate moment problems; 22. The Riemann–Hilbert problem; 23. Multiple orthogonal polynomials; 24. Research problems; Bibliography; Index.