Taylor coefficients of non-holomorphic Jacobi forms and applications

TL;DR

This paper demonstrates that Taylor coefficients of certain non-holomorphic Jacobi forms possess modular properties, extending known results from holomorphic cases and exploring their specific spaces in two examples.

Contribution

It proves modularity of Taylor coefficients of non-holomorphic Jacobi forms and identifies their precise functional spaces in specific cases.

Findings

Taylor coefficients of non-holomorphic Jacobi forms are modular

Identified the functional spaces for these coefficients in two examples

Extended modularity results beyond holomorphic Jacobi forms

Abstract

In this paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well-known that Taylor coefficients of holomorphic Jacobi forms are quasimoular forms. However recently there has been a wide interest for Taylor coefficients of non-holomorphic Jacobi forms for example arising in combinatorics. In this paper, we show that such coefficients still inherit modular properties. We then work out the precise spaces in which these coefficients lie for two examples.