Polynomial Generalizations of Rogers-Ramanujan Type Identities

The Rogers-Ramanujan identities are a pair of analytic/formal power series identities, each of which assert the equality of a certain infinite series with an infinite product. They were first discovered by the English mathematician L.J. Rogers in 1894. Later, it was realized that the series and products could be viewed as generating functions for certain classes of integer partitions, and thus the Rogers-Ramanujan identities are also combinatorial identities. Around 1980, Australian physicist Rodney Baxter showed that the Rogers-Ramanujan identities were intimately linked to the solution of the hard hexagon model in statistical mechanics. Around the same time, Rutgers mathematicians Jim Lepowsky and Robert Wilson gave the first Lie theoretic interpration and proof of the Rogers-Ramanujan identities. This work eventually led to the discovery of vertex operator algebras.

In my talk, I plan to give a brief but motivating introduction to q-series, and then denomonstrate an elementary method by which any q-series/infinite product identity of Rogers-Ramanujan type can be generalized to a polynomial identity, which in turn has important implications in statistical physics and algorithmic proof theory.