The Eichler integral of $E_2$ and $q$-brackets of $t$-hook functions

For functions $f: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, Bloch and Okounkov defined a power series $\langle f\rangle_q$ that is the "weighted average" of $f$. As Fourier series in $q=e^{2πi z}$, such $q$-brackets generate the ring of quasimodular forms, and the modular forms that are powers of Dedekind's eta-function. Using work of Berndt and Han, we build modular objects from $f_t(λ):= t\sum_{h\in \mathcal{H}_t(λ)}\frac{1}{h^2},$ weighted sums over partition hook numbers that are multiples of $t$. We find that $\langle f_t \rangle_q$ is the Eichler integral of $(1-E_2(tz))/24,$ which we modify to construct a function $M_t(z)$ that enjoys weight 0 modularity properties. As a consequence, the non-modular Fourier series $H_t^*(z):=\sum_{λ\in \mathcal{P}} f_t(λ)q^{|λ|-\frac{1}{24}}$ inherits weight $-1/2$ modularity properties. These are sufficient to imply a Chowla-Selberg type result, generalizing the fact that weight $k$ algebraic modular forms evaluated at discriminant $D<0$ points $τ$ are algebraic multiples of $Ω_D^k,$ the $k$th power of the canonical period. If we let $Ψ(τ):=-πi \left(\frac{τ^2-3τ+1}{12τ}\right)-\frac{\log(τ)}{2},$ then for $t=1$ we prove that $H_1^*(-1/τ)-\frac{1}{\sqrt{-iτ}}\cdot H_1^*(τ)\in \overline{\mathbb{Q}}\cdot \frac{Ψ(τ)}{\sqrt{Ω_D}}.$

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