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

We consider the $t$-hook functions on partitions $f_{a,t}: \mathcal{P}\rightarrow \mathbb{C}$ defined by $f_{a,t}(λ):=t^{a-1} \sum_{h\in \mathcal{H}_t(λ)}\frac{1}{h^a},$ where $\mathcal{H}_t(λ)$ is the multiset of partition hook numbers that are multiples of $t$. The Bloch-Okounkov $q$-brackets $\langle f_{a,t}\rangle_q$ include Eichler integrals of the classical Eisenstein series. For even $a\geq 2$, we show that these $q$-brackets are natural pieces of weight $2-a$ sesquiharmonic and harmonic Maass forms, while for odd $a\leq -1,$ 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.

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