On the Fourier expansion of Bloch-Okounkov

TL;DR

This paper investigates the algebraic and analytic properties of Fourier coefficients of the Bloch-Okounkov n-point function, revealing their relation to Rogers' false theta function and deriving their asymptotic behavior.

Contribution

It provides new results on the Fourier coefficients of the Bloch-Okounkov function, including their asymptotics and connections to false theta functions, and introduces higher rank generalizations.

Findings

Fourier coefficients relate to Rogers' false theta function.

Asymptotic behavior of coefficients as τ approaches 0 is established.

Higher rank generalizations of Bloch-Okounkov functions are introduced.

Abstract

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 $\tau \rightarrow 0$, 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.