Rank-Crank type PDEs and generalized Lambert series identities

TL;DR

This paper derives Rank-Crank PDEs for higher order Appell functions from generalized Lambert series identities, unifying and extending previous results and proofs using elliptic functions.

Contribution

It introduces a unified approach to derive Rank-Crank PDEs from generalized Lambert series identities, extending classical results and proofs.

Findings

Derived Rank-Crank PDEs from generalized Lambert series identities

Connected special PDEs to classical Lambert series identities by Watson and Jackson

Extended Atkin and Swinnerton-Dyer's elliptic function proof to generalized identities

Abstract

We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson respectively. The first author's Lambert series identities are common generalizations. We also show how Atkin and Swinnerton-Dyer's proof using elliptic functions can be extended to prove these generalized Lambert series identities.