Paraphrasing the authors, the book under review (hereafter BW2) is a “considerable enlargement” of their *Special Functions* (hereafter BW1), which I have previously reviewed here. I will try not to repeat myself too much. I compared BW1 to the book of the same title by Andrews, Askey, and Roy (hereafter AAR), also published by Cambridge University Press, which grew out of a graduate course at the University of Wisconsin that I was fortunate to take several times. The comparison was generally to the detriment of BW1, and I am now inclined to think that I overdid this a little. I would still use AAR if I were to teach a graduate course in special functions, but BW1 is a fine book, BW2 is better, and it is a closer decision now.

As advertised, BW2 has quite a bit more material than BW1 even though it is only 17 pages longer. It is a tiny bit taller and has a lower bottom margin, but the difference is due mainly to the elimination of the chapter summaries that occupied 89 pages of BW1. The authors admit that this was done partly to save space, but also remark that these “often proved to be more annoying than helpful in use of the book for reference.” Call me a snob if you want, but I don’t think a book of this quality should have chapter summaries. Whittaker and Watson doesn’t have chapter summaries.

BW2 has four new chapters. The old chapter 4 has been split into chapters 4 and 5, and the new chapter 4 has a very good concise summary of the general theory of orthogonal polynomials. BW2 is much more competitive with AAR on this topic than BW1 and has some advantages. One is chapter 4’s proof of Favard’s theorem on the orthogonality of polynomial sequences satisfying a three-term recurrence relation. Another is the new chapter 7, an account of recent work of Wong and some coauthors on a new approach to the asymptotics of orthogonal polynomials, with the Hermite polynomial case worked out in detail. These two chapters justify the addition of “orthogonal polynomials” to the title of BW2, although I wonder if chapter 7 might be better placed after chapter 13.

Many readers will find it easy to decide between BW2 and AAR, one way or the other, because the two books take very different points of view. AAR puts hypergeometric functions at the center of the subject, while second order differential equations are fundamental for Beals and Wong. 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 \(x^n/n!\) can be written as a quotient of products of shifted factorials \((a)_n=a (a+1) \dots (a+n-1)\), where \((a)_0=1\). If there are \(p\) of these in the numerator and \(q\) of them in the denominator, then we have a \({}_pF_q\) 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 \(e^x\), and a \({}_1F_0\) is just the binomial series for \((1-x)^{-a}\). Besides these, the most important case is the \({}_2F_1\), which satisfies a second order differential equation with three regular singular points, usually taken at \(0\), \(1\), and \(\infty\). This equation specializes to the ones satisfied by Hermite, Laguerre, Legendre, and Jacobi polynomials. A hypergeometric function is an analytic continuation of a hypergeometric series.

One of the areas where I was most critical of BW1 was its decision to ignore hypergeometric functions except for the \({}_2F_1\) and the \({}_1F_1\), which they denote instead by \(F\) and \(M\) respectively. This is only slightly improved in BW2, where the first two sections of the new chapter 12 discuss the general \({}_pF_q\) as an entry to the Meijer \(G\)-function. Readers of the 2013 article by Beals and Szmigielski in the *Notices of the American Mathematical Society *(vol. 60, pp. 866–872) will have expected to see this material in BW2, and it is another of the book’s advantages over AAR. Some material on orthogonal polynomials of more general hypergeometric type — *e.g.,* the continuous Hahn polynomials, which are \({}_3F_2\)s discovered in 1985 and were one of my favorite polynomial sequences in graduate school — has also been added to chapter 6 (formerly chapter 5). The contrast with AAR, in which (for example) section 3.4 presents Whipple’s transformation relating a *very well poised* \({}_7F_6\) to a *balanced * \({}_4F_3\), remains stark. This is one of the most surprising results in special functions, and many important summation formulas flow from it.

The concluding chapter 15 is also new. It treats Painlevé transcendents, an unusual topic for a special functions book. They do not appear in the two classic handbooks, the Bateman manuscript project *Higher Transcendental Functions *and Abramowitz and Stegun’s *Handbook of Mathematical Functions*, but the newer *NIST Handbook of Mathematical Functions *devotes chapter 32 to them. Because they are solutions of (nonlinear) second order differential equations having a special property, Painlevé transcendents fit in well with the authors’ approach (better than the previous chapter on elliptic functions), and as these functions find more and more applications in physics, random matrix theory, differential geometry, and elsewhere, the decision to include them may soon look prescient.

By far the most common approach to special functions is through second order differential equations, but the treatment of Beals and Wong is distinctive. One might, for example, begin a chapter on Hermite polynomials by writing down the Schrödinger equation for the quantum mechanical harmonic oscillator, which after some substitutions takes the form \(d^2u/dz^2 = (z^2-2E)u\). (Here \(E\) is an energy level, and turns out to be half an odd integer.) This has an elegant solution via raising and lowering operators (due to Dirac? If anyone knows for sure, please write to me): \(u\) turns out to be essentially a Gaussian times a Hermite polynomial whose degree comes from \(E\), and one can then develop the properties of Hermite polynomials from this point of view.

This is not what Beals and Wong do. (I was describing something I did in a mathematical methods of physics course.) Already on page 2 they ask when a second order linear homogeneous differential equation with analytic coefficients would have an analytic solution with a nice recursive structure. After a page of calculations, they find that this happens essentially only when we have the hypergeometric equation (satisfied by the \({}_2F_1\)) or the confluent hypergeometric equation (satisfied by the \({}_1F_1\)). They are very much concerned with trying to explain, from a pure rather than an applied perspective, why the special functions are special.

Except as noted above, BW2 is almost unchanged from BW1. The reader may refer to my earlier review for some of the things that BW2 may be said to be missing. Chapter 1 gives a brief overview of what it is not missing. Chapter 2 studies the gamma and beta functions, which, although unrelated to differential equations, are indispensable for later calculations. For example, the beta function is itself a combination of three gamma functions, and Gauss found the sum of a convergent \({}_2F_1\) when \(x=1\) in terms of four gamma functions. A more modern topic is the Selberg integral, a far-reaching extension of the beta function, where the authors give Aomoto’s beautiful proof and extension. They also give a brief account of the Riemann zeta function at this point, both for its intrinsic interest and because the functional equation involves the gamma function.

Chapter 3 presents some general theory of second order differential equations. The authors make frequent use of what they call *gauge transformations, *in which the dependent variable \(u(x)\) is replaced by \(v(x)g(x)\) for a new dependent variable \(v\) and a nonzero \(g\) to recast the equations, *e.g.,* to eliminate the first derivative term. This chapter also has more discussion of the specialness of the special functions. In section 3.4 the authors ask a natural question about polynomial eigenfunctions of second order differential operators and find that the only possibilities are Hermite, Laguerre, and Jacobi polynomials. Section 3.6 introduces the standard technique of separation of variables for partial differential equations.

As mentioned above, chapter 4 treats the general theory of orthogonal polynomials and is mostly new. Most of the old chapter 4 is now chapter 5, which discusses the classical orthogonal polynomials. These are the ones mentioned in the previous paragraph and also Legendre and Chebyshev polynomials, which are special cases of Jacobi polynomials but have enough distinctive properties to warrant additional development. The remarks at the end of this chapter contain the authors’ final statement about why these polynomials are special.

Chapter 6 is now titled Semi-classical orthogonal polynomials. These are mostly discrete, *i.e., *the orthogonality relation is a sum instead of an integral (the title of the old chapter 5 was Discrete orthogonal polynomials), but there is a new section on some less well known continuous sets, such as the continuous Hahn polynomials mentioned above. Still missing are \(q\)-orthogonal polynomials, which are mentioned only in the remarks for chapters 5 and 6.

BW2 has two chapters on asymptotics, the new chapter 7 and chapter 13, which used to be chapter 10. Chapter 13 does not refer back to chapter 7 even when it treats Hermite polynomials. As I observed in my earlier review, the exercises on asymptotics tend to be much longer (or at least to take much longer to state) than those in the other chapters. Wong is one of the world’s leading experts on asymptotics, and this is another area where BW2 can claim an advantage over AAR. One could go further in this direction by using Olver’s classic *Asymptotics and Special Functions*, but BW2 is more up to date.

To one used to hypergeometric functions, it seems strange to have chapter 10 on the \({}_2F_1\) before chapter 8 on the \({}_1F_1\), since a \({}_1F_1\) comes from a \({}_2F_1\) by replacing \(x\) by \(x/b\), where \(b\) is one of the two numerator parameters, and then letting \(b\) tend to \(\infty\) (in fact, this is exercise 1 in chapter 10), but this is not illogical from the differential equations point of view. Chapter 9 is on cylinder functions, by which the authors mostly mean Bessel functions. Neither AAR nor BW2 go into much depth here, though both are fine as far as they go. The same can be said of BW2 on hypergeometric functions, where anyone seriously interested in the topic would certainly do better to read chapters 2 and 3 of AAR.

Still missing from BW2 is the Pfaff-Saalschütz identity for terminating \({}_3F_2\) series of a certain kind, a corollary of Pfaff’s transformation on page 257. Many binomial coefficient identities are special cases of the Pfaff-Saalschütz identity, but the authors have no interest in combinatorics. This also shows up in exercise 6.17, where the authors give a hint involving partial derivatives, when one has only to use the fact that \(k\) times \(\binom{n}{k}\) equals \(n\) times \(\binom{n-1}{k-1}\). Both sides count the number of ways to choose a team of \(k\) players, including a captain, from a pool of \(n\) players.

Chapter 11 is a nice short treatment of spherical harmonics, a subject in which Legendre functions appear almost everywhere. The last of the old chapters is on elliptic functions. The only \(q\)-analysis in the book is here, in sections 14.4 and 14.5 on theta functions.

The omission of \(q\)-analysis (my favorite part of the subject) is mostly down to the point of view that Beals and Wong have taken. At least so far, \(q\)-orthogonal polynomials have been of much less practical importance than the classical \(q=1\) case. They are solutions of difference equations rather than differential equations, and play a crucial role in the classification problem for orthogonal polynomials. But \(q\)-hypergeometric series, mentioned here only in the remarks for chapter 10, are even more interesting than ordinary ones, so it would be unconscionable to leave them out of a treatment where the focus was on hypergeometric functions.

The book also has two short appendices, one on complex analysis and one on Fourier analysis. Each chapter concludes with a section of Exercises and then some historical and bibliographical remarks. There are 436 exercises in all, 88 more than BW1. Most of these fill gaps in the exposition, but some stand alone. One of the latter that jumped out at me was exercise 5.14, a striking equality of two integrals for which it is hard to imagine an application. The hint is fine; the parameter in the proposed substitution should really be \(s\) instead of \(t\) to be consistent with the rest of it, but anyone capable of doing the other steps would surely figure this out. The historical and bibliographical remarks occupy a total of about 13 1/2 pages and are quite well done.

BW2 is an excellent graduate textbook, one of the two best available on this subject, with enough differences in content and motivation from AAR that some readers may prefer it. BW2 is an analyst’s special functions book, while AAR is a little more broad.

Warren Johnson ([email protected]) is Associate Professor of Mathematics at Connecticut College. He is trying to finish a book on \(q\)-analysis.