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

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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 Qₖ, and study the p-adic properties of their q-brackets. To do this, we define regularized versions Qₖ⁽ᵖ⁾ for primes p. We also use Jacobi forms to show that the <Qₖ⁽ᵖ⁾>q are quasimodular and find explicit expressions for them in terms of the <Qₖ>q.