\(q\)-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra.

*(English)*Zbl 0594.33001
Regional Conference Series in Mathematics 66. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0716-1). xii, 130 p. (1986).

Every once in a while a book is published that is so beautiful that one rejoices to be around to read it. The present book is one such. The subject is basic hypergeometric series, or \(q\)-series. These are series on the form \(\sum a_ n\) with \(a_{n+1}/a_ n\) a rational function of \(n\), and some multidimensional extensions. This seems an unpromising topic, yet there are beautiful applications in combinatorics, number theory, statistical mechanics, orthogonal polynomials, Lie and Kac-Moody algebras, group theory, and a marvelous illustration of the power of symbolic algebra on a computer, in this case SCRATCHPAD.

\(q\)-series start with Euler, but the modern period really starts in England, first with Sylvester’s work on partitions, and second with a very important series of papers by L. J. Rogers. The work of Rogers was completely ignored until one of his papers was read by Ramanujan twenty years after it was written. The main series of three papers of Rogers have been read seriously by a few people, but the paper that led to this trilogy probably had not been read carefully before Andrews read it, and redid it in Chapter 2. Andrews than goes along the same path Rogers took, but uses SCRATCHPAD to obtain enough data to explain to the reader how this material could have been discovered. After Ramanujan’s rediscovery of the Rogers-Ramanujan identities, and his reading of Rogers’s paper on them, new proofs of these identities were found by Rogers and Ramanujan. Bailey had read one of the later papers of Rogers carefully, and found a simple method of forming new series from old. This is explained, and illustrated by obtaining Rogers-Ramanujan identities for mod \((2k+1)\), \(k=2,3,....\)

Integrals and series are clearly analogous to each other. There are times when there is much more than an analogy. Using the orthogonality of \(e^{in\theta}\) it is possible to change many integrals into sums, and vice versa. One very important instance of this is the set of integrals, or constant term identities, that start with Selberg’s multidimensional beta integral, includes the Dyson-Gunson-Wilson identity, the extension of it proved by Zeilberger and Bressoud, and then up to the Macdonald conjectures. Part of this is treated, including the first inclusion of Selberg’s integral with a proof in a book. Since Selberg’s paper is not readily obtainable, this is a service to the many who can use this integral.

The other start of a modern treatment of \(q\)-series is Sylvester’s work on partitions, with the aim of providing bijective proofs of identities, or showing that two sets are equinumerous by providing an explicit bijection. Some beautiful bijections of Bressoud are given, as is a proof of the very important involution principle of Garsia and Milne. Next, generalized Frobenius partitions are introduced, and are shown to be a very powerful tool to handle certain questions.

The application of most interest outside of pure mathematics is Baxter’s solution of the hard hexagon model in two dimensional statistical mechanics. This is a fascinating story, and the end is not in sight. Andrews, Baxter, Forrester and SCRATCHPAD have done some marvelous work, adding many more exactly solved models to those previously solved by Onsager, Baxter and a few others. A small part of this work is outlined here. A most tantalizing set of functions was introduced by Ramanujan in his last letter to Hardy. He called these mock theta functions, and gave examples of some of orders 3, 5 and 7. Watson explained some of the theory of the third order ones in two papers in the mid 1930’s, and wrote a bit about the fifth order ones. There is still a lot we do not know about these functions, but Andrews with SCRATCHPAD has made substantial progress. The book ends with a description of how to use SCRATCHPAD to attack problems in this area. First, Andrews shows how the Lusztig, Macdonald, Wall conjecture could have been solved using SCRATCHPAD. This is much easier than the original proof by Andrews. Bailey’s work along with the desire to understand the mock theta functions led Andrews to infinitely many identities, with the mock theta functions on the second level. On the first, or easiest, level, he found the remarkable identity; \[ (\sum^{\infty}_{n=0}q^{n(n+1)/2})^ 3=\sum^{\infty}_{n=0}\sum^{2n}_{j=0}q^{2n^ 2+2n- j(j+1)/2}(1+q^{2n+1})(1-q^{2n+1})^{-1}. \] This gives a new proof of Gauß’s theorem that every positive integer is the sum of three triangular numbers. If you have not found something of beauty that you like by the end of this book, you have my sympathies. I found a lot.

\(q\)-series start with Euler, but the modern period really starts in England, first with Sylvester’s work on partitions, and second with a very important series of papers by L. J. Rogers. The work of Rogers was completely ignored until one of his papers was read by Ramanujan twenty years after it was written. The main series of three papers of Rogers have been read seriously by a few people, but the paper that led to this trilogy probably had not been read carefully before Andrews read it, and redid it in Chapter 2. Andrews than goes along the same path Rogers took, but uses SCRATCHPAD to obtain enough data to explain to the reader how this material could have been discovered. After Ramanujan’s rediscovery of the Rogers-Ramanujan identities, and his reading of Rogers’s paper on them, new proofs of these identities were found by Rogers and Ramanujan. Bailey had read one of the later papers of Rogers carefully, and found a simple method of forming new series from old. This is explained, and illustrated by obtaining Rogers-Ramanujan identities for mod \((2k+1)\), \(k=2,3,....\)

Integrals and series are clearly analogous to each other. There are times when there is much more than an analogy. Using the orthogonality of \(e^{in\theta}\) it is possible to change many integrals into sums, and vice versa. One very important instance of this is the set of integrals, or constant term identities, that start with Selberg’s multidimensional beta integral, includes the Dyson-Gunson-Wilson identity, the extension of it proved by Zeilberger and Bressoud, and then up to the Macdonald conjectures. Part of this is treated, including the first inclusion of Selberg’s integral with a proof in a book. Since Selberg’s paper is not readily obtainable, this is a service to the many who can use this integral.

The other start of a modern treatment of \(q\)-series is Sylvester’s work on partitions, with the aim of providing bijective proofs of identities, or showing that two sets are equinumerous by providing an explicit bijection. Some beautiful bijections of Bressoud are given, as is a proof of the very important involution principle of Garsia and Milne. Next, generalized Frobenius partitions are introduced, and are shown to be a very powerful tool to handle certain questions.

The application of most interest outside of pure mathematics is Baxter’s solution of the hard hexagon model in two dimensional statistical mechanics. This is a fascinating story, and the end is not in sight. Andrews, Baxter, Forrester and SCRATCHPAD have done some marvelous work, adding many more exactly solved models to those previously solved by Onsager, Baxter and a few others. A small part of this work is outlined here. A most tantalizing set of functions was introduced by Ramanujan in his last letter to Hardy. He called these mock theta functions, and gave examples of some of orders 3, 5 and 7. Watson explained some of the theory of the third order ones in two papers in the mid 1930’s, and wrote a bit about the fifth order ones. There is still a lot we do not know about these functions, but Andrews with SCRATCHPAD has made substantial progress. The book ends with a description of how to use SCRATCHPAD to attack problems in this area. First, Andrews shows how the Lusztig, Macdonald, Wall conjecture could have been solved using SCRATCHPAD. This is much easier than the original proof by Andrews. Bailey’s work along with the desire to understand the mock theta functions led Andrews to infinitely many identities, with the mock theta functions on the second level. On the first, or easiest, level, he found the remarkable identity; \[ (\sum^{\infty}_{n=0}q^{n(n+1)/2})^ 3=\sum^{\infty}_{n=0}\sum^{2n}_{j=0}q^{2n^ 2+2n- j(j+1)/2}(1+q^{2n+1})(1-q^{2n+1})^{-1}. \] This gives a new proof of Gauß’s theorem that every positive integer is the sum of three triangular numbers. If you have not found something of beauty that you like by the end of this book, you have my sympathies. I found a lot.

Reviewer: R. Askey

##### MSC:

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |

05A19 | Combinatorial identities, bijective combinatorics |

11P84 | Partition identities; identities of Rogers-Ramanujan type |

68W30 | Symbolic computation and algebraic computation |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

11F37 | Forms of half-integer weight; nonholomorphic modular forms |