A theorem of G\"ollnitz and its place in the theory of partitions

ABSTRACT

A Rogers-Ramanujan (R-R) type identity is a q-hypergeometric identity which is in the form of a series equal to a product, with the series representing the generating function of partitions whose parts satisfy
difference  conditions and the product being the generating function of partitions whose parts satisfy congruence conditions. R-R type identities arise in a variety of settings ranging from the study of vertex operators in Lie algebras to exactly solvable models in physics. One of the deepest
R-R type identities is a 1967 theorem of G\"ollnitz. We will describe a new approach to the G\"ollnitz theorem using the combinatorics of words and view the theorem as emerging from an incredible  three parameter q-hypergeometric identity. As a consequence we get combinatorial insights into Jacobi's triple product identity for theta functions and certain partition congruences modulo powers of 2. Companion results to G\"ollnitz's theorem can be constructed as well. We will also briefly indicate what lies beyond G\"ollnitz's theorem in four free parameters.
The talk will include joint work with George Andrews, Basil Gordon,
and Alexander Berkovich.