  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 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.
