Partial Theta Functions. I. Beyond the Lost Notebook

It is shown how many of the partial theta function identitiesin Ramanujan's lost notebook can be generalized to infinitefamilies of such identities. Key in our construction is theBailey lemma and a new generalization of the Jacobi triple productidentity. By computing residues around the poles of our identitieswe find a surprising connection between partial theta functionidentities and Garrett–Ismail–Stanton-type extensionsof multisum Rogers–Ramanujan identities. 2000 MathematicsSubject Classification 05A30, 33D15, 33D90.