On the Fourier expansion of Bloch-Okounkov n-point function

In this paper, we study algebraic and analytic properties of Fourier coefficients, expressed as q-series, of the so-called Bloch-Okounkov n-point function. We prove several results about these series and explain how they relate to Rogers' false theta function. Then we obtain their full asymptotics, as τ→0, and use this result to derive asymptotic properties of the coefficients in the q-expansion. At the end, we also introduce and discuss higher rank generalization of Bloch-Okounkov's functions.