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

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 q-brackets \(\left _q\)) 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 \(\left _q\) are quasimodular and find explicit expressions for them in terms of the \(\left _q\).