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

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.

Automated summary

This article explores families of functions on partitions, specifically shifted symmetric functions, and their corresponding q-brackets as quasimodular forms. By extending these families, we are able to obtain quasimodular q-brackets for a congruence subgroup. Additionally, we identify certain subspaces within these families where the q-brackets are modular forms. These findings are based on the properties of Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.