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On the Fourier coefficients of negative index meromorphic Jacobi forms
In this paper, we consider the Fourier coefficients of meromorphic Jacobi
forms of negative index. This extends recent work of Creutzig and the first two
authors for the special case of Kac-Wakimoto characters which occur naturally
in Lie theory, and yields, as easy corollaries, many important PDEs arising in
combinatorics such as the famous rank-crank PDE of Atkin and Garvan. Moreover,
we discuss the relation of our results to partial theta functions and quantum
modular forms as introducted by Zagier, which together with previous work on
positive index meromorphic Jacobi forms illuminates the general structure of
the Fourier coefficients of meromorphic Jacobi forms.Comment: 13 pages, minor change
Higher Order SPT-Functions
Andrews' spt-function can be written as the difference between the second
symmetrized crank and rank moment functions. Using the machinery of Bailey
pairs a combinatorial interpretation is given for the difference between higher
order symmetrized crank and rank moment functions. This implies an inequality
between crank and rank moments that was only know previously for sufficiently
large n and fixed order. This combinatorial interpretation is in terms of a
weighted sum of partitions. A number of congruences for higher order
spt-functions are derived.Comment: 21 pages (previous version was 19 pages), added reference to Andrews
and Rose's recent paper, MacMahon's paper and OEIS, changed some wordin
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