Reciprocals of false theta functions

Abstract

We investigate reciprocals of false theta functions, producing results such as congruences, simple asymptotic bounds, and combinatorial identities. Of particular interest is a connection between $1/\hat{I}¨(-q^2,q)$ and the truncated pentagonal number theorem of Andrews and Merca. We record a useful dissection identity analogous to the known theta function dissection.

Submitted for Proceedings of the Legacy of Ramanujan Conference 2024, in honor of the 85th birthdays of George Andrews and Bruce Berndt. v2: Minor typographical corrections and changes made from reader comments. No mathematical change