Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities

TL;DR

This paper generalizes five of Ramanujan's lesser-known q-series identities, deriving new identities and weighted partition identities through analytic methods, expanding understanding of Ramanujan's work and partition theory.

Contribution

It provides a unified one-variable generalization of Ramanujan's five identities and introduces new weighted partition identities derived from these generalizations.

Findings

Derived a single general identity from which all five Ramanujan identities follow.

Established new weighted partition identities from the generalized identities.

Provided an analytic proof for a generalization of Bressoud and Subbarao's identity.

Abstract

Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities, but didn't establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from which we successfully deduce all the five q-series identities of Ramanujan. In addition to this, we also establish a few interesting weighted partition identities from our generalized identity. In the mid 1980's, Bressoud and Subbarao found an interesting identity connecting the generalized divisor function with a weighted partition function, which they proved by means of a purely combinatorial argument. Quite surprisingly, we found an analytic proof for a generalization of the identity of Bressoud and Subbarao, starting from the fourth identity of the aforementioned five q-series identities of Ramanujan.