Rank-Crank type PDE's and non-holomorphic Jacobi forms

TL;DR

This paper explores the connection between Rank-Crank PDEs and non-holomorphic Jacobi forms, deriving an infinite family of differential equations and applying them to establish new congruences for odd Durfee symbols.

Contribution

It introduces a framework linking Rank-Crank PDEs with non-holomorphic Jacobi forms and derives an infinite family of such differential equations.

Findings

Derived an infinite family of Rank-Crank PDEs within non-holomorphic Jacobi forms framework.

Established new congruences for odd Durfee symbols using these PDEs.

Connected classical partition statistics with modern automorphic forms.

Abstract

In this paper we show how Rank-Crank type PDE's (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews.