When is the Bloch\u2013Okounkov q-bracket\xa0modular?

Abstract

We obtain a condition describing when the quasimodular forms given by the Bloch–Okounkov theorem as q-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator Delta . We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.