Generalizations of Ramanujan's reciprocity theorem and their applications

Abstract

First, we briefly survey Ramanujan's reciprocity theorem for a certain q-series related to partial theta functions and give a new proof of the theorem. Next, we derive generalizations of the reciprocity theorem that are also generalizations of the 1ψ1 summation formula and Jacobi triple product identity and show that these reciprocity theorems lead to generalizations of the quintuple product identity, as well. Last, we present some applications of the generalized reciprocity theorems and product identities, including new representations for generating functions for sums of six squares and those for overpartitions.