# Closed form expression for $\psi_{e^{\pi}}^{(3)}(1)$

Let $$\psi_q(x)$$ be the q-DiGamma function defined for a real variable $$x>0$$ as $$\psi_q(x)=\frac{1}{\Gamma_q(x)}\frac{\partial}{\partial x} (\Gamma_q(x))$$ where $$\Gamma_q(x)$$ is the q-Gamma function defined as $$\Gamma_q(x)=(1-q)^{1-x}\prod_{n=0}^{\infty}\frac{1-q^{n+1}}{1-q^{n+x}}$$

Question I am looking for a closed form for $$\psi_{e^{\pi}}^{(3)}(1)$$

If anyone could please solve this by hand or mathematica or sage math. I would be highly indebted to you all.

Wolfram Alpha gives the expansion at $$x=\infty$$:

$$\psi_x^{(3)}(1)=\ln^4(x)\left(x^{-1}+9x^{-2}+\dots\right)$$

and these match Oeis A$$001158$$ with divisor $$\sigma_v(n)$$ and various theta functions after plugging the sum back in here. Use $$\vartheta_v(0,x)=\vartheta_v(x)$$:

$$\psi_x^{(3)}(1)=\ln^4(x)\sum_{n=1}^\infty\frac{\sigma_3(n)}{x^n}=\frac{\ln^4(x)}{480}\left(\vartheta_2\left(\frac1{\sqrt x}\right)^8+ \vartheta_3\left(\frac1{\sqrt x}\right)^8+ \vartheta_4\left(\frac1{\sqrt x}\right)^8-2\right)$$

Therefore:

$$\psi_{e^\pi}^{(3)}(1)=\frac{\pi^4}{480}\left(\vartheta_2^8\left(e^{-\frac\pi2}\right)+ \vartheta_3^8\left(e^{-\frac\pi2}\right)+ \vartheta_4^8\left(e^{-\frac\pi2}\right)-2\right)$$

Clicking “more digits” here shows a smaller error each time implying the result is true.

Now use Dedekind $$\eta(z)$$ identities for $$\vartheta_v\left(e^{-\frac\pi2}\right)$$ when $$v=2$$, $$v=3$$, and $$v=4$$

$$\psi_{e^\pi}^{(3)}(1)= \frac{\pi^4}{480}\left(\left(2\frac{\eta^2(i)}{\eta\left(\frac i2\right)}\right)^8+\left(\frac{\eta^5\left(\frac i2\right)}{\eta^2\left(i\right)\eta^2\left(\frac i4\right)}\right)^8+\left(\frac{\eta^2\left(\frac i4\right)}{\eta\left(\frac i2\right)}\right)^8-2\right)$$

Using special values in terms of $$\Gamma\left(\frac14\right)$$:

$$\eta\left(\frac i4\right)=2\eta(4i)=\frac{\sqrt[4]{\sqrt2-1} \Gamma\left(\frac14\right)}{2^\frac{13}{16}\pi^\frac34},\eta\left(\frac i2\right)=\frac{\Gamma\left(\frac14\right)}{2^\frac 78\pi^\frac34},\eta(i)=\frac{\Gamma\left(\frac14\right)}{2\pi^\frac34}$$

Finally, substitute and have a form in terms of $$\Gamma\left(\frac14\right)$$ which has no elementary closed form. Therefore:

$$\boxed{\psi_{e^\pi}^{(3)}(1)=\frac{11\Gamma\left(\frac14\right)^8}{5120\pi^2}-\frac{\pi^4}{240}}$$

shown here

• I am indebted to you for your effort. Thank you. I would really appreciate if a closed form can be found of $\vartheta_v\left(e^{-\frac\pi2}\right)$
– Max
Commented Feb 18, 2023 at 18:02
• @Anixx Thank you for your edit. One small help please. The link of Dedekind eta function $\eta(z)$ is not working
– Max
Commented Feb 19, 2023 at 1:22