The Bloch-Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms

TL;DR

This paper extends the theory of q-brackets of functions on partitions to congruence subgroups, showing they are quasimodular or modular, based on properties of Taylor coefficients of quasi-Jacobi forms.

Contribution

It generalizes the Bloch-Okounkov theorem to congruence subgroups and links q-brackets to Taylor coefficients of quasi-Jacobi forms.

Findings

q-brackets are quasimodular for congruence subgroups

certain families yield modular q-brackets

Taylor coefficients characterize modularity properties

Abstract

There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets are quasimodular forms. We extend these families so that the corresponding q-brackets are quasimodular for a congruence subgroup. Moreover, we find subspaces of these families for which the q-bracket is a modular form. These results follow from the properties of Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.