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Special Functions: A Graduate Text

Richard Beals and Roderick Wong
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 126
[Reviewed by
Warren Johnson
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Perhaps the most striking omission (there are several) from the book under review (hereafter BW) is an explanation of why it was written. The authors are, or were, outsiders to the special functions community: Beals is an outstanding analyst, now Professor Emeritus at Yale; Wong is an expert on the kindred subject of asymptotics at the City University of Hong Kong. On his web site Beals modestly says “Through various research projects I was drawn, reluctantly, into a passing acquaintance with special functions,” but in fact BW demonstrates a mastery of the classical part of the subject.

It is also not clear to me why Cambridge University Press chose to publish BW, as they brought out a book with the same title by Andrews, Askey, and Roy (hereafter AAR) eleven years earlier, which is still in print. AAR is in the Encyclopedia of Mathematics and its Applications series and BW in the Studies in Advanced Mathematics series, but both are graduate textbooks. BW says as much in a subtitle, while AAR grew out of a graduate course at the University of Wisconsin that I was fortunate to take several times. (I’m trying to be objective, but the reader should bear this in mind.) It is hard to see BW as anything other than a competitor of AAR. At least CUP has the high-end textbook market in special functions cornered.

One of the distinctive features of AAR is the decision to put hypergeometric series at the heart of the subject, whereas many other books on special functions emphasize certain second order differential equations that arise in mathematical physics. A hypergeometric series is just a power series where the ratio of successive coefficients is a rational function of the summation index. This implies that the coefficient of xn/n! can be written as a quotient of products of shifted factorials (a)n=a(a+1)…(a+n–1), where (a)0=1. If there are p of these in the numerator and q of them in the denominator, then we have a pFq hypergeometric series. Many elementary functions can be written in this form for small values of p and q. For example, when p=q=0 we have the series for ex, and a 1F0 is just the binomial series for (1–x)–a. Besides these, the most important case is the 2F1, which satisfies a second order differential equation with three regular singular points, usually taken at 0, 1, and ∞. This equation specializes to the ones satisfied by Hermite, Laguerre, Legendre, and Jacobi polynomials.

Beals and Wong put second order differential equations back into the center of the subject, but the approach is more theoretical than phenomenological. (It seems appropriate to borrow this distinction from the physicists here.) BW begins by asking when a second order differential equation with analytic coefficients would have a solution that is analytic and has a nice recursive structure. The answer is that this happens essentially only when we have the hypergeometric equation (satisfied by the 2F1) or the confluent hypergeometric equation (satisfied by the 1F1).

Although it is an old subject, special functions has had a renaissance in my lifetime. The celebrated WZ algorithm of Wilf and Zeilberger now furnishes automatic proofs of most hypergeometric series identities. The classical orthogonal polynomials have arisen as matrix elements in the representation theory of Lie groups, and other functions arising in representation theory have come to be regarded as special. Some of these are q-analogues of the classical orthogonal polynomials, and we now have a whole theory of q-special functions (including q-gamma and q-beta functions) largely parallel to the classical theory. Another parallel theory of elliptic special functions (cf. Eric Rains’s talk at the joint meetings in January 2012) is still in the early stages of construction.

We also understand general orthogonal polynomials much better than we used to. For example, we have found all the hypergeometric and q-hypergeometric orthogonal polynomials of one variable, and we have combinatorial interpretations of many of them. Although primarily focused on the classical theory, AAR has much more q-analysis than BW, and hints of these other developments. BW is a fine book, but I believe it is a step backward from AAR.

Chapter 1 of BW gives a brief overview. As I mentioned above, it attempts, in sections 1.1 and 1.3, to explain what is special about the special functions, from the point of view of second order differential equations. Chapter 2 studies the gamma, beta, and zeta functions, where differential equations are not fruitful. It includes a section on the Selberg integral (with Aomoto’s beautiful proof), a modern topic that gets a whole chapter in AAR. Another nice feature in this chapter is Sasvari’s lovely argument for Binet’s formula for the gamma function, from which one can get improvements of Stirling’s formula. Chapter 3 presents some general theory of second order differential equations. The authors make frequent use of what they call gauge transformations, a fancy name for changing the dependent variable. This is preparation for chapter 4 (the longest of the eleven chapters) on the classical orthogonal polynomials. Chapter 5 is on discrete orthogonal polynomials, another more modern topic, but the q-orthogonal polynomials, which are mostly discrete, are mentioned only in one paragraph at the end of the Remarks section of chapter 4.

Chapter 6 is on confluent hypergeometric functions. It seems odd to see these before general hypergeometric functions (though these are mentioned briefly in chapters 1 and 3), but this is not illogical from the differential equations point of view. Chapter 7 is on cylinder functions, mostly Bessel functions. Here is a topic where AAR leaves one wanting more, but BW is not an improvement.

We finally reach hypergeometric functions in chapter 8 — in AAR they are in chapters 2 and 3. BW treats the 2F1 almost exclusively, though the pFq is mentioned briefly, and the 1F1 was in chapter 6. This allows BW to suppress the subscripts 2 and 1 in most of this chapter, as in many older books, but it also gives up a lot. For example, one of the most interesting results in the whole of special functions is Whipple’s transformation, which relates a very well-poised 7F6 to a balanced 4F3.

Beals and Wong derive Pfaff’s transformation for the 2F1, which has two main applications: (i) a more beautiful 2F1 transformation of Euler; and (ii) the very useful Pfaff-Saalschütz identity — many binomial coefficient identities are special cases — for terminating 3F2 series of a certain kind. BW cannot even mention (ii), let alone Whipple’s transformation, because it does not discuss pFq series with q>1 or p>2. One could argue that the Pfaff-Saalschütz identity is less interesting now that we have the WZ method, but BW doesn’t mention it either, except in passing in the Remarks section of this chapter.

This lack of interest in binomial identities also shows up in exercise 5.16, where the authors give a hint involving partial derivatives, when one has only to use the fact that k times {n choose k} equals n times {n–1 choose k–1}. Both sides count the number of ways to choose a team of k players, including a captain, from a pool of n players.

Chapter 9 is on spherical harmonics and Legendre functions. Some of the asymptotics of the special functions are discussed where they arise, but the rest are deferred to chapter 10. Chapter 11 is an introduction to elliptic functions that could have been written in the 19th century. The only q-series in BW are here, in connection with theta functions. There are also two short appendices, one on complex analysis and one on Fourier analysis.

Each chapter has several sections of text (the mean is about 6.7), followed by some exercises, a summary, and some historical and bibliographical remarks. There are 348 exercises in all, many of which fill gaps in the exposition. The exercises run to 13 pages in the chapter on asymptotics and an average of 4 pages in the other chapters, which probably tells us something about the two authors. The exercises in chapter 10 are not more numerous, just more expansive: exercise 10.21 occupies as much space as the 26 exercises in chapter 6.

The chapter summaries are a distinctive feature of BW. They restate the most important formulas in each chapter with minimal commentary. The ratio of summary to text (i.e., the sections preceding the exercises) is about .34. It surprises me that Cambridge University Press would sanction 89 pages of redundancy, but that’s between them and the authors. Some readers will surely appreciate it.

The historical and bibliographical remarks run to about 10 pages, 2 in chapter 4 (including the authors’ final statement about why the classical orthogonal polynomials are special) and ½ or 1 in the other chapters. These are valuable, although clearly influenced by AAR. BW is remarkably old-fashioned, but few mathematicians today are as old-fashioned as I am, so coming from me that’s not really an insult. While it is not as good as AAR, it is a solid graduate textbook. An instructor who wants to emphasize differential equations and is not interested in q-analysis might prefer it, and it is at least as good on asymptotics as AAR.

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

Preface; 1. Orientation; 2. Gamma, beta, zeta; 3. Second order differential equations; 4. Orthogonal polynomials; 5. Discrete orthogonal polynomials; 6. Confluent hypergeometric functions; 7. Cylinder functions; 8. Hypergeometric functions; 9. Spherical functions; 10. Asymptotics; 11. Elliptic functions; References; Index.