Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions

Abstract

We consider the $t$-hook functions on partitions $f_{a,t}:\mathcal{P}\to \mathbb{C}$ defined by $$f_{a,t}(\lambda ):=t^{a-1}\sum _{h\in {\mathcal{H}}_t(\lambda )}\frac{1}{h^a},$$ where ${\mathcal{H}}_t(\lambda )$ is the multiset of partition hook numbers that are multiples of $t$. The Bloch-Okounkov $q$-brackets $⟨f_{a,t}{⟩}_q$ include Eichler integrals of the classical Eisenstein series. For even $a\ge 2$, we show that these $q$-brackets are natural pieces of weight $2-a$ sesquiharmonic and harmonic Maass forms, while for odd $a\le -1,$ we show that they are holomorphic quantum modular forms. We use these results to obtain new formulas of Chowla-Selberg type, and asymptotic expansions involving values of the Riemann zeta-function and Bernoulli numbers. We make use of work of Berndt, Han and Ji, and Zagier.

This paper has been submitted to a volume in memory of Dick Askey. This version corrects typos and a sign mistake identified by the referee