George E. Andrews

Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

AbstractThe eight-vertex model is equivalent to a “solid-on-solid” (SOS) model, in which an integer heightli is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1⩽li⩽4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1⩽li⩽r−1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponentsβ, α,

Applications of Basic Hypergeometric Functions

\bar \alpha

Problems and Prospects for Basic Hypergeometric Functions

are obtained.

Enumeration of Partitions: The Role of Eulerian Series and q-Orthogonal Polynomials

holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6) in the same way that (1.1) and (1.2) treat the primes 5 and 7. In [4, 5], one of us was able to find a crank relative to vector partitions as follows: For a partition 7r, let #(7r) be the number of parts of ir and cr{n) be the sum of the parts of ir (or the number ir is partitioning) with the convention #( ) = ) = 0 for the empty partition 0, of 0. Let

The number of smallest parts in the partitions of n

This paper surveys recent applications of basic hypergeometric functions to partitions, number theory, finite vector spaces, combinatorial identities and physics.

Arithmetic properties of partitions with even parts distinct

In the spring of 1976, the first author visited Trinity College Library at Cambridge University. Dr. Lucy Slater had suggested to him that there were materials deposited there from the estate of the late G.N. Watson that might be of interest to him. In one box of materials from Watson’s estate, Andrews found several items written by Srinivasa Ramanujan. The most interesting item in this box was a manuscript written on 138 sides in Ramanujan’s distinctive handwriting. The sheets contained over six hundred mathematical formulas without proofs. Although technically not a notebook, and although technically not “lost,” as we shall see in the sequel, it was natural in view of the fame of Ramanujan’s (earlier) notebooks [5] to call this manuscript Ramanujan’s lost notebook. Almost surely, this manuscript, or at least most of it, was written during the last year of Ramanujan’s life, after his return to India from England. We do not possess a bona fide proof of this claim, but we shall later present considerable evidence for it. The manuscript contains no introduction or covering letter. In fact, there are hardly any words in the manuscript. There are a few marks evidently made by a cataloguer, and there are also a few remarks in the handwriting of G.H. Hardy. Undoubtedly, the most famous objects examined in the lost notebook are the mock theta functions , about which more will be said later. Concerning this manuscript, Ms. Rosemary Graham, manuscript cataloguer of the Trinity College Library, remarked, “. . . the notebook and other material was discovered among Watson’s papers by Dr. J.M. Whittaker, who wrote the obituary of Professor Watson for the Royal Society. He passed the papers to Professor R.A. Rankin of Glasgow University, who, in December 1968, offered them to Trinity College so that they might join the other Ramanujan manuscripts already given to us by Professor Rankin on behalf of Professor Watson’s widow.” Since her late husband had been a fellow and scholar at Trinity College and had had an abiding, lifelong affection for Trinity College, Mrs. Watson agreed with Rankin’s suggestion that the library at Trinity College would be the best place to preserve her husband’s papers. Since Ramanujan had also been a fellow at Trinity College, Rankin’s suggestion was even more appropriate. The natural, burning question now is, How did this manuscript of Ramanujan come into Watson’s possession? We think that the manuscript’s history can be traced.

Lattice gas generalization of the hard hexagon model. III.q-Trinomial coefficients

Publisher Summary This chapter discusses certain problems and prospects for basic hypergeometric functions. It also focuses on finite linear homogeneous ordinary q-difference equations with coefficients that are polynomials in x and q, which have multiple basic hypergeometric series as solutions. The chapter also presents possible multiple series generalizations of the q-analog of Whipples theorem. It also focuses on whether there are multiple series q-analogs of well-poised hypergeometric series that specialize to the cases of the quituple product identity, or Winquists identity or some other multiple theta series that sum to an infinite product. The chapter further reviews MacMahons master theorem and the Dyson conjecture.

MacMahon's Partition Analysis

The theory of partitions has long been associated with so called basic hypergeometric functions or Eulerian series. We begin with discussion of some of the lesser known identities of L.J. Rogers which have interesting interpretations in the theory of partitions. Illustrations are given for the numerous ways partition studies lead to Eulerian series. The main portion of our work is primarily an introduction to recent work on orthogonal polynomials defined by basic hypergeometric series and to the applications that can be made of these results to the theory of partitions. Perhaps it is most interesting to note that we deduce the Rogers-Ramanujan identities from our solution to the connection coefficient problem for the little q-Jacobi polynomials.