The smallest parts function associated with $\omega(q)$

TL;DR

This paper proves new congruences modulo powers of 5 for Fourier coefficients related to mock theta functions and smallest parts functions, confirming some conjectures and expanding understanding of these special functions.

Contribution

It establishes new congruences for smallest parts functions associated with mock theta functions, linking Fourier coefficients to partition theory and confirming conjectures.

Findings

Proved congruences modulo powers of 5 for Fourier coefficients.

Established congruences for two smallest parts functions.

Confirmed two conjectural congruences of Wang.

Abstract

We establish two families of congruences modulo powers of 5 for the Fourier coefficients of $(2E_2(2\tau)-E_2(\tau))\eta(2\tau)^{-1}$, where $E_2(\tau)$ is the weight 2 Eisenstein series and $\eta(\tau)$ is the Dedekind eta function. This allows us to prove similar congruences for two smallest parts functions. The first function is $\mathrm{spt}_{\omega}(n)$, which was introduced by Andrews, Dixit and Yee and associated with Ramanujan/Watson's third order mock theta function $\omega(q)$. The second one is $\mathrm{spt}_{C5}(n)$, which appeared in the work of Garvan and Jennings-Shaffer. Moreover, we confirm two conjectural congruences of Wang.