Partial theta functions. I. Beyond the lost notebook

TL;DR

This paper generalizes many of Ramanujan's partial theta function identities using the Bailey lemma and a new Jacobi triple product generalization, revealing unexpected links to multisum Rogers-Ramanujan identities.

Contribution

It introduces a novel approach to extend partial theta identities beyond Ramanujan's lost notebook using advanced q-series techniques.

Findings

Many identities are generalized to infinite families.

A new connection to multisum Rogers-Ramanujan identities is established.

Residue calculations reveal surprising links between different q-series identities.

Abstract

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta functions identities and Garret-Ismail-Stanton-type extensions of multisum Rogers-Ramanujan identities.