Reciprocals of false theta functions

TL;DR

This paper explores the reciprocals of false theta functions, revealing new congruences, asymptotic bounds, combinatorial identities, and connections to classical theorems like the pentagonal number theorem.

Contribution

It introduces novel identities and dissection formulas for reciprocals of false theta functions, linking them to established results in number theory.

Findings

Derived congruences for reciprocals of false theta functions

Established asymptotic bounds for these reciprocals

Connected reciprocals to the truncated pentagonal number theorem

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 $1/\Psi(-q^2,q)$ and the truncated pentagonal number theorem of Andrews and Merca. We record a useful dissection identity analogous to the known theta function dissection.