Series Representation of Gamma Function

Question

The $\Gamma(x)$ is function That has derivatives in the polygamma form. Can those derivatives be used to make a Taylor series? I've tried but I got stuck as soon as I find out That $\Psi^1(1)=\zeta(2)=\frac{\pi^2}{6}$ and that $\Psi_0^2(1)=(-\gamma) ^2$

I am doing the series at $x_0=1$ and $\zeta(z)$ is the Reimann Zeta Function and $\gamma$ is the Euler-Mascheroni constant.

Answer 1

The derivatives of the gamma function are a pain.

Better ways to compute it are Stirling's approximation or the Lanczos approximation as described here:

https://en.m.wikipedia.org/wiki/Lanczos_approximation