The Eichler Integral of E 2 and q-brackets of t-hook Functions

For functions \(f: \mathcal {P}\mapsto \mathbb {C}\) on partitions, Bloch and Okounkov defined a power series 〈f〉q that is the “weighted average” of f. As Fourier in q = e2πiz, such q-brackets generate ring quasimodular forms, modular forms are powers Dedekind’s eta-function. Using work Berndt Han, we build objects from $$\displaystyle f_t(\lambda ):= t\sum _{h\in {H}_t(\lambda )}\frac {1}{h^2}, $$ weighted sums over partition hook numbers multiples t. We find 〈ft〉q Eichler integral (1 − E2(tz))∕24, which modify to construct function Mt(z) enjoys weight 0 modularity properties. consequence, non-modular H_t^*(z):=\sum _{\lambda \in {P}} )q^{|\lambda |-\frac {1}{24}} $$ inherits 1∕2 These sufficient imply Chowla-Selberg type result, generalizing fact k algebraic evaluated at discriminant D < points τ \(\Omega _D^k,\) kth canonical period. If let \(\Psi (\tau ):=-\pi i \left (\frac {\tau ^2-3\tau +1}{12\tau }\right )-\frac {\log )}{2},\) then for t 1 prove \begin{aligned} H_1^*(-1/\tau {1}{\sqrt {-i\tau }}\cdot H_1^*(\tau )\in \overline {\mathbb {Q}}\cdot \frac {\Psi )}{\sqrt {\Omega _D}}. \end{aligned} $$