Modularity of Taylor coefficients of non-holomorphic Jacobi forms arising from partitions

초록

It is well-known that Taylor coefficient of holomorphic Jacobi forms is quasimodular forms, and recently it was proved by Bringmann that such a property still holds for certain non-holomorphic Jacobi forms arising in combinatorics. In this paper, we prove further modularity results of an infinite family of combinatorial sums related to non-holomorphic Jacobi forms. More precisely, for each k≥2 and ℓ≥1 we find non-holomorphic modular forms r2ℓ−1,k(τ) arising from partitions. It turns out that the non-holomorphic part of r1,k(τ) is related to the generalized Rogers-Ramanujan functions. This appearance of generalized Rogers-Ramanujan functions is supported by the fact that this r1,k(τ) is a mixed harmonic Maass form. © 2024 Elsevier Inc.

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