Rank-Crank type PDEs and generalized Lambert series identities

Details Rank-Crank type PDEs and generalized Lambert series identities Rank-Crank type PDEs and generalized Lambert series identities

2012-01-08

arxiv.org/abs/1201.1663v1

Mathematics

Number Theory

Scientific paper

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.

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