Several new product identities in relation to two-variable Rogers-Ramanujan type sums and mock theta functions

TL;DR

This paper discovers new product identities involving two-variable Rogers-Ramanujan sums and mock theta functions, revealing deep connections between classical identities and modern mock theta functions through a novel algebraic framework.

Contribution

It introduces a new perspective on product-sum identities as representations in vector spaces of quasiperiodic functions, and establishes novel correspondences linking classical identities with mock theta functions.

Findings

New identities for nonuple, undecuple, and two-variable Rogers-Ramanujan sums.

A main theorem linking septuple product identity with Rogers-Ramanujan identities via mock theta functions.

Applications to identities involving generalized Dedekind eta functions and mock theta functions.

Abstract

Product identities in two variables $x, q$ expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers-Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers-Ramanujan identities, involving two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers-Ramanujan identities, which is related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized Dedekind eta functions and mock theta functions.