Congruences modulo powers of $5$ and $7$ for the crank and rank parity functions and related mock theta functions

TL;DR

This paper extends known congruences modulo powers of 5 and 7 for the crank and rank parity functions, and relates these to mock theta functions using modular transformations and involutions.

Contribution

It introduces new congruences for the rank and crank parity functions modulo powers of 7 and connects these to mock theta functions via modular transformations.

Findings

Congruences modulo powers of 7 for the rank parity function are established.

New congruences for the crank parity function modulo powers of 7 are proved.

Relations between these congruences and mock theta functions are demonstrated.

Abstract

It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of $5$ for the crank parity function. The generating function for the analogous rank parity function is $f(q)$, the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. Recently we proved congruences modulo powers of $5$ for the rank parity function, and here we extend these congruences for powers of $7$. We also show how these congruences imply congruences modulo powers of $5$ and $7$ for the coefficients of the related third order mock theta function $\omega(q)$, using Atkin-Lehner involutions and transformation results of Zwegers. Finally we a prove a family of congruences modulo powers of $7$ for the crank parity function.