Multi-dimensional $q$-summations and multi-colored partitions

TL;DR

This paper presents a combinatorial proof of a multi-dimensional overpartition identity that generalizes classical results like Cauchy's identity and theta function product formulas, extending Sylvester's classical identity.

Contribution

It introduces a new overpartition analogue with parameters for overlined parts of different colors, unifying several classical identities within a single framework.

Findings

Provides a combinatorial proof of the new identity.

Encompasses classical results as special cases.

Extends Sylvester's identity to multi-dimensional overpartitions.

Abstract

Motivated by Alladi's recent multi-dimensional generalization of Sylvester's classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy's identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.