Parity of coefficients of mock theta functions

TL;DR

This paper investigates the parity properties of coefficients in classical mock theta functions, classifying their distribution of even and odd coefficients, and conjectures about their parity types.

Contribution

It classifies 21 out of 44 mock theta functions as having coefficients mostly even, and conjectures the parity types for the remaining functions, providing new insights into their coefficient behavior.

Findings

21 mock theta functions are of parity type (1,0)

Conjectures on parity types for remaining functions

Characterizations of n with odd coefficients for certain functions

Abstract

We study the parity of coefficients of classical mock theta functions. Suppose $g$ is a formal power series with integer coefficients, and let $c(g;n)$ be the coefficient of $q^n$ in its series expansion. We say that $g$ is of parity type $(a,1-a)$ if $c(g;n)$ takes even values with probability $a$ for $n\geq 0$. We show that among the 44 classical mock theta functions, 21 of them are of parity type $(1,0)$. We further conjecture that 19 mock theta functions are of parity type $(\frac{1}{2},\frac{1}{2})$ and 4 functions are of parity type $(\frac{3}{4},\frac{1}{4})$. We also give characterizations of $n$ such that $c(g;n)$ is odd for the mock theta functions of parity type $(1,0)$.