A Bijective Proof of a False Theta Function Identity from Ramanujan’s Lost Notebook

Abstract

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