Special Functions

This is a very erudite text and reference in special functions, that concentrates on functions defined by hypergeometric series. “We have decided to focus primarily on the best understood class of functions, hypergeometric functions, and the associated hypergeometric series.” (p. xiv) The erudition consists of (1) drawing connections and analogies between many areas of mathematics; (2) giving many surprising applications, also drawn from many areas of mathematics (number theory and combinatorics are especially well-represented).

 

The topics covered in most detail are the gamma and beta functions, hypergeometric series, and orthogonal polynomials. There’s an especially good chapter on q-series, that does an excellent job of motivating them.  There are shorter chapters on partitions and on Bailey chains. There are several concise appendices that explain the many methods from classical analysis that are used in the book. The main omission is most of the functions of mathematical physics (unless they happen to be hypergeometric); elliptic functions are covered very briefly. Asymptotics of the functions studied is covered only in a few cases.

 

Apart from that, this is a thorough treatment of the topics it covers, that focuses on proofs (often giving several of the same theorem); it is not a mere listing of definitions and properties. Each chapter has numerous exercises; these are a mix of proving things “left to the reader”, and developing interesting topics that are peripheral to the main narrative.  There’s a concise description of the Gosper and the Wilf–Zeilberger methods of mechanical summation of hypergeometric terms; these have often been found useful in proving hypegeometric identities.

 

There is thorough coverage of orthogonal polynomials, divided into three chapters. The first chapter deals with common properties of all sequences of orthogonal polynomials, based on the three-term recurrence that all such sequences satisfy. It includes a digression showing the analogies with continued fractions. The second chapter defines particular sequences and studies their properties, including expansions in orthogonal functions. The third chapter deals primarily with Jacobi polynomials considered as hypergeometric functions. It includes a proof of Apéry’s result that the Riemann zeta function value \( \zeta(3) \) is irrational, an application of Legendre polynomials. This result fascinates me and I’m always happy when a book includes it.

 

The writing is very clear and generally easy to follow. Each chapter starts with a fairly lengthy introduction, that not only previews the subject matter but explains how it relates to other subjects.

 

One thing I especially like about this book is how it occasionally jumps from the continuous world to examine analogs in the discrete or finite-field worlds. For example, there’s a splendid discussion of how the gamma and beta functions have finite-field analogs as the Gauss and Jacobi sums from number theory, which is followed far enough to prove Fermat’s theorem that every prime of the form 4n + 1 is the sum of two squares.

 

Now I will look at some competitors of the present book (hereafter abbreviated AAR). The NIST Handbook of Mathematical Functions is an indispensable reference for special functions, but it does not give any proofs, or much in the way of explanations, so it does not compete directly with AAR. The NIST Handbook does give all the definitions and the most important facts and formulas and is thoroughly footnoted so you can look up the proofs if you need them. This book seems to be out of print in print format, but the information is freely available and continues to be updated regularly at the NIST Digital Library of Mathematical Functions website.

 

Traditionally the “one to beat” in this field was Whittaker and Watson’s A Course of Modern Analysis, which was described by Alf van der Poorten as “the bible of the classical special functions”. The first half of the book is a complete course in classical analysis, not especially related to special functions. The second half, titled “Transcendental Functions”, is the bible. Like AAR, Whittaker and Watson gives complete proofs, and has excellent exercises. AAR is also about the classical special functions (both books have a distinct nineteenth-century flavor at times), but the emphasis and selection is very different. Whittaker and Watson’s organizing principle is complex analysis and power series rather than hypergeometric series. They also emphasize the differential equations of mathematical physics. As hinted by the title of the second part, this book does not deal with orthogonal polynomials, except in Chapter 15 on Legendre functions. There’s not a great deal of overlap in the two books; Whittaker and Watson generally go into more detail on the topics they have in common.

 

Another book, that I have not seen but got a very favorable review in MAA Reviews, is Beals and Wong’s Special Functions and Orthogonal Polynomials. This one also concentrates on the differential equations of mathematical physics, but within this specialization is very complete and up-to-date (published in 2017, versus 1999 for AAR).

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Allen Stenger is a math hobbyist and retired software developer.  His personal website is allenstenger.com. His mathematical interests are number theory and classical analysis.