abstract of Andrew Sills (q2020)

Speaker

Andrew Sills (Georgia Southern University)

Date

October 6, 8:30-9:30, Japan (October 5, 19:30-20:30, Georgia, USA)

Title

A SURVEY OF ROGERS--RAMANUJAN TYPE IDENTITIES

Abstract

In 1894, L. J. Rogers published a paper on the expansion of certain infinite products that included a pair of q-series identities that were later rediscovered independently by S. Ramanujan (at first without proof). These analytic identities, traditionally written in the variable “q”, each asserted the equality of an infinite product with an infinite series where the general term involved a power of q and rising q-factorials. Ramanujan's mentor G. H. Hardy was unaware of Rogers' work, and circulated the identities among various European mathematicians as unproven conjectures. Later Ramanujan discovered a proof, and then came across Rogers' earlier work in an old issue of the *Proceedings of the London Mathematical Society*. These identities came to be known as the Rogers--Ramanujan identities. In hindsight, some earlier identities due to Euler, Jacobi, and Heine could be classified as identities “of Rogers--Ramanujan type.” Rogers and Ramanujan each discovered quite a few identities of similar type in the 1910s, but then the subject remained relatively dormant until W. N. Bailey (who knew Ramanujan while undergraduate at Cambridge) and his then-Ph.D. student L. J. Slater revived the study of Rogers--Ramanujan type identities in the 1940s. These identities of analytic functions can also be viewed as generating functions for classes of combinatorial objects (often integer partitions). This combinatorial view became particularly popular in the 1960s under the leadership of Basil Gordon and George Andrews, when a torrent of infinite families of combinatorial identities of Rogers--Ramanujan type were discovered. Since the 1980s, many additional applications of Rogers--Ramanujan type identities have been found in a variety of areas of mathematics and physics.

In this talk, I will outline the history of the subject, with an emphasis on the perspective of basic hypergeometric series and integer partitions. Recent advances and open problems will be discussed at the end as time allows.