Andrews-Beck Type Congruences Related to the Crank of a Partition

TL;DR

This paper explores recent conjectures about the crank of a partition, analyzing generating functions and connections to mock theta functions, advancing understanding of partition ranks and cranks.

Contribution

It revisits conjectures on the crank of a partition, decomposes generating functions, and links them to mock theta functions and series, providing new insights.

Findings

Decomposition of crank generating functions

Connections established with Apple-Lerch series

Relations to tenth order mock theta functions

Abstract

In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a suitable generating function, while the conjectures related to cranks were studied by Shane Chern using weighted partition moments. We revisit the conjectures on the crank of a partition by decomposing the relevant generating function and further explore connections with Apple-Lerch series and tenth order mock theta functions.