Bressoud-Subbarao type weighted partition identities for a generalized divisor function

TL;DR

This paper revisits and generalizes Bressoud-Subbarao weighted partition identities using combinatorial and analytic methods, introduces new identities via fractional differential operators, and extends related identities involving Bell polynomials.

Contribution

It provides a more general form of Bressoud-Subbarao identities, introduces new weighted partition identities using fractional calculus, and generalizes Uchimura's identity with Bell polynomials.

Findings

Derived a more general weighted partition identity.

Established new identities using fractional differential operators.

Extended Uchimura's identity to a one-variable case.

Abstract

In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a $q$-series identity of Ramanujan. In the present paper, we revisit the combinatorial arguments of Bressoud and Subbarao, and derive a more general weighted partition identity. Furthermore, with the help of a fractional differential operator, we establish a few more Bressoud-Subbarao type weighted partition identities beginning from an identity of Andrews, Garvan and Liang. We also found a one-variable generalization of an identity of Uchimura related to Bell polynomials.