On spt-crank type functions

TL;DR

This paper investigates the asymptotic behavior and sign patterns of a new spt-type function related to Ramanujan's mock theta functions, confirming a positivity conjecture and exploring congruences.

Contribution

It analyzes the asymptotics and sign patterns of a recently introduced spt-type function and confirms a related positivity conjecture.

Findings

Confirmed the positivity conjecture asymptotically.

Analyzed the sign pattern of the crank function.

Studied congruences for the spt-type function.

Abstract

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function $\operatorname{spt}_\omega(n)$, which is closely related to Ramanujan's third order mock theta function $\omega(q)$. Garvan and Jennings-Shaffer introduce a crank function which explains congruences for $\operatorname{spt}_\omega(n)$. In this note, we study asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for $\operatorname{spt}_\omega(n)$.