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