On a conjecture of Berndt and Kim

TL;DR

This paper proves a recent conjecture by Berndt and Kim about the positivity of coefficients in the asymptotic expansion of certain partial theta functions, extending classical and recent results in the field.

Contribution

It establishes the positivity conjecture for a broad class of partial theta functions, generalizing previous specific cases and connecting to Ramanujan's work.

Findings

Confirmed positivity of coefficients in the asymptotic expansion

Extended classical results to a more general setting

Linked modern conjecture with Ramanujan's classical work

Abstract

We prove a recent conjecture of Berndt and Kim regarding the positivity of the coefficients in the asymptotic expansion of a class of partial theta functions. This generalizes results found in Ramanujan's second notebook, and recent work of Galway and Stanley.