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We consider the $t$-hook functions on partitions $f_{a,t}:\mathcal{P}\rightarrow \mathbb{C}$ 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 aremultiples of $t$. The Bloch-Okounkov $q$-brackets $\langle f_{a,t}\rangle_q$include Eichler integrals of the classical Eisenstein series. For even $a\geq2$, 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 thatthey are holomorphic quantum modular forms. We use these results to obtain newformulas of Chowla-Selberg type, and asymptotic expansions involving values ofthe Riemann zeta-function and Bernoulli numbers. We make use of work of Berndt,Han and Ji, and Zagier.

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