Partial theta functions. I. Beyond the lost notebook

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.