Topics and Methods in q-Series

I recently finished teaching a senior seminar dealing with an introduction to the theory of modular forms, and accordingly revisited some of Srinivasa Ramanujan’s magic. Specifically, late in the course we had occasion to take a (necessarily very cursory) look at his mock-theta functions. This was preceded by a more serious look at theta-functions properly so-called, and in that context we travelled back to the 18th century, and encountered Leonhard Euler, who, as Dunham puts it so well, is the master of us all. It’s reasonable to bracket Euler and Ramanujan together, of course, given the particular fecundity of their work, its frequent algorithmic character (for lack of a better word), and the sheer virtuosity of their handling certain mathematical fauna, series among them. And for us number theorists, these two titans are among our very favorites: they’re on everyone’s list.

The book under review is concerned with a theme going back to Euler, and championed by Ramanujan in connection with, e.g., the aforementioned mock-theta functions, namely, \(q\)-series. The book ends with a chapter on these beasts; it (sort of) starts with a chapter on the \(q\)-binomial theorem. To get an idea of what we’re playing with in this unique part of analytic number theory, here’s what this binomial theorem looks like: \[\sum_0^\infty \frac{(a;q)_n}{(q;q)_n}z^n = \frac{(az;q)_\infty}{(z;q)_\infty}.\] One is moved to respond, with Bertie Wooster, “eh, what?” Well, we clearly need to know the definition of the symbol \((a;q)_n\), and it’s not so bad: \((a;q)_n=\displaystyle\prod_0^{n-1}(1-aq^j)\), and now it’s crystal-clear … but, as George Andrews puts it in his Foreword to this book, “intense.” And then Andrews, certainly one of the subject’s gurus, goes on to say, encouragingly, “Don’t be put off by the size of some of the formulas … [There are] a number of symmetries and surprising special cases that, with familiarity, become captivating.”

Indeed, I guess it’s fair to say that, as with so many things in mathematics when the gloves come off, a large part of one’s investment in learning things consists in getting comfortable a new language: just think of what needs to be done when learning, e.g., homological algebra, what with all those marvelous arrows finessed to give everything from commutative diagrams to exact sequences — the long and short of it so to speak. And it’s clear that with \(q\)-series, it’s the same sort of thing. Let’s just say that familiarity breeds, not contempt, but comfort.

On to the book proper, then. Let me start by saying that it’s a very good one. McLaughlin covers the subject along a historical trajectory, hitting a large number of marvelous themes. We encounter the already mentioned -binomial theorem at the start, and then go on to, e.g., Jacobi’s triple product identity (in which context we meet Ramanujan again), Bailey pairs and chains, \(q\)-continued fractions, Lambert series, and finally the mock-thetas of Ramanujan. McLaughlin provides seven appendices, six of which are by way of overviews of what’s available. For example, Appendix VI, dealing with mock theta functions, is a long list of identities of various orders involving the mock thetas, and we naturally once again encounter Ramanujan, now often accompanied by G. H. Hardy. A propos, speaking of Hardy and acknowledging that this subject of \(q\)-series is not generally part of boot camp and training of today’s arithmeticians, I want to mention what is probably the premier source of classical number theory in its broad sense, viz., Hardy and Wright’s Introduction to the Theory of Numbers. Although you won’t find \(q\)-series here in any explicit way, reading this classic book is a fantastic way to get into the right frame of mind for going out on the mathematical town with Ramanujan and Hardy.

In closing, let me also note that Topics and Methods in \(q\)-Series, in addition to presenting such a wonderful sweep of deep and beautiful material, is very strong pedagogically. It’s very well written, in an accessible and clear style, the material dealt with is effectively motivated and discussed in a sound and rigorous manner, all the proofs are there, and McLaughlin gives the reader a large number of exercises to do along the way, as he travels these paths. Again, the expressions and formulas that pepper the pages of this book are at first sight daunting, but they do yield to persistent pressure, as per Andrews’ observation cited above. The language of \(q\)-series makes for some beautiful poetry.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.