Combinatorial Proofs of an Identity from Ramanujan's Lost Notebook and its Variations

TL;DR

This paper provides two independent combinatorial proofs and interpretations of a Ramanujan identity from his lost notebook, extending previous work and employing involution principles and bijections from partition theory.

Contribution

It introduces new combinatorial proofs and interpretations of a Ramanujan identity, including a direct involution-based proof and a bijective proof via partition theory generalization.

Findings

Two independent combinatorial proofs of the identity

Extension of the identity to related cases

A bijective proof using generalized partition bijections

Abstract

We examine an identity originally stated in Ramanujan's ``lost notebook'' and first proven algebraically by Andrews and combinatorially by Kim. We give two independent combinatorial proofs and interpretations of this identity, which also extends an identity recently proven by Pak and Waarnar related to the product of partial theta functions: First, we give a direct combinatorial proof, using the involution principle, of a special case of the identity, and extend this into a direct combinatorial proof of the full identity as written. Second, we show that the identity can be rewritten, using minor algebraic manipulation, into an identity that can be proven with a direct bijection. We provide such a bijection using a generalization of a standard bijection from partition theory.