Partition implications of a three-parameter q-series identity

Abstract

A generalization of a beautiful q-series identity found in the unorganized portion of Ramanujan’s second and third notebooks is obtained. As a consequence, we derive a three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews et al., which itself generalizes the famous result of Fokkink et al. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural proof gives interesting number-theoretic information along the way. We also obtain a new result consisting of an infinite series involving a special case of Fine’s function F(a, b;t), namely F(0, qn ; cqn). For c = 1, this gives Andrews’ famous identity for spt(n), whereas for c = −1, 0 and q, it unravels new relations that the divisor function d(n) has with other partition-theoretic functions such as the largest parts function lpt(n).