# On Series Expansions of Capparelli's Infinite Product

## TL;DR

This paper introduces two new series representations for the infinite product linked to Capparelli's Rogers-Ramanujan type identity, expanding the analytic understanding of these combinatorial identities and presenting related new identities.

## Contribution

It provides novel series expansions for Capparelli's infinite product and introduces additional related identities, enhancing the analytic framework of Rogers-Ramanujan type identities.

## Key findings

- Two new series representations for Capparelli's infinite product
- Additional related identities and infinite families presented
- Enhanced analytic understanding of Capparelli's conjecture

## Abstract

Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers Ph.D. thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof. Most combinatorial Rogers-Ramanujan type identities (e.g. the G\"ollnitz-Gordon identities, Gordon's combinatorial generalization of the Rogers-Ramanujan identities, etc.) have an analytic counterpart. The main purpose of this paper is to provide two new series representations for the infinite product associated with Capparelli's conjecture. Some additional related identities, including new infinite families are also presented.