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

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 < f >(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 < Q(k)((p))>(q) are quasimodular and find explicit expressions for them in terms of the < Q(k)>(q).