Does anyone know closed form expressions for $$\eta^{(k)}(i)$$ up to high $k \in \mathbf{N}$? ($\eta$ is the Dedekind eta function.) For instance, I can use Mathematica to obtain $$\eta(i) = \frac{\Gamma \left(\frac{1}{4}\right)}{2 \pi ^{3/4}}$$ $$\eta'(i) = \frac{1}{12} i e^{-25 \pi /12} \left(24 \pi \text{QPochhammer}^{(0,1)}\left(e^{-2 \pi },e^{-2 \pi }\right)\\ \qquad + e^{2 \pi } \left(e^{-2 \pi };e^{-2 \pi }\right){}_{\infty } \left(12 \left(\psi _{e^{-2 \pi }}^{(0)}(1)+\log \left(1-e^{-2 \pi }\right)\right)+\pi \right)\right) $$

but going higher than $k=3$ takes too long. Additionally, Mathematica also has some trouble numerically evaluating the derivatives of QPochhammers and QPolyGammas that appear.

If a closed form is not available, tips for rapidly evaluating to very high precision the values of $\eta^{(k)}(i)$ would also be greatly appreciated.

  • 2
    $\begingroup$ $\log \eta(z)$ has a simple expression from which you can compute its derivatives. There is the operator $f \in M_k(SL_2(\Bbb{Z})) \to \vartheta_k f(z)= \frac{f'(z)}{2i\pi} - \frac{k}{12} E_2(z)f(z) \in M_k(SL_2(\Bbb{Z}))$ where $E_2(z) = \frac{1}{48 i \pi} \frac{\eta'(z)}{\eta(z)}$ $\endgroup$ – reuns Jun 6 at 21:55

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