On p-adic modular forms and the Bloch-Okounkov theorem

TL;DR

This paper investigates the $p$-adic properties of $q$-brackets associated with shifted symmetric polynomials, extending the Bloch-Okounkov theorem by introducing regularized forms and connecting to Jacobi forms.

Contribution

It introduces regularized $Q_k^{(p)}$ functions for primes $p$, analyzes their $q$-brackets, and establishes their quasimodularity using Jacobi forms, extending the original theorem.

Findings

$Q_k^{(p)}$ $q$-brackets are quasimodular forms

Explicit relations between regularized and original $q$-brackets

Connection to Jacobi forms for proving quasimodularity

Abstract

Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left<f\right>_q$) are quasimodular forms. We revisit a family of such functions, denoted $Q_k$, and study the $p$-adic properties of their $q$-brackets. To do this, we define regularized versions $Q_k^{(p)}$ for primes $p.$ We also use Jacobi forms to show that the $\left<Q_k^{(p)}\right>_q$ are quasimodular and find explicit expressions for them in terms of the $\left<Q_k\right>_q$.