Rank-Crank type PDE's for higher level Appell functions

TL;DR

This paper derives partial differential equations for higher level Appell functions, generalizing known Rank-Crank PDEs for specific levels, thereby expanding the understanding of their mathematical structure.

Contribution

It introduces a unified PDE framework for all odd levels of Appell functions, extending previous results for levels 3 and 5.

Findings

Derived PDEs for all odd levels of Appell functions

Unified approach generalizing Rank-Crank PDEs

Confirmed known PDEs for levels 3 and 5

Abstract

In this paper we consider level l Appell functions, and find a partial differential equation for all odd l. For l=3 this recovers the Rank-Crank PDE, found by Atkin and Garvan, and for l=5 we get a similar PDE found by Garvan.