Author： Chan Song
Publisher： Springer Publishing Company
ISSN： 1382-4090
Source： The Ramanujan Journal, Vol.31, Iss.1-2, 2013-06, pp. : 163-189
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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 identity is a common generalization. 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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