A bijective proof of a false theta function identity from Ramanujan's lost notebook

TL;DR

This paper provides a new combinatorial proof and interpretation of a false theta function identity from Ramanujan's lost notebook, introducing multivariate generalizations and extending understanding of these identities.

Contribution

It offers the first bijective proof of a specific false theta identity and develops multivariate generalizations using novel combinatorial methods.

Findings

New combinatorial interpretation of Ramanujan's false theta identity

Bijective proof establishing the identity rigorously

Multivariate generalizations of the original identity

Abstract

In his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}.$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtained a combinatorial interpretation of another one of the identities.