Looking at what the python scipy library does for its special functions, the polygamma is found by returning the digamma if the zeroth derivative is requested, otherwise return $(-1)^{n+1}\Gamma(n+1)\zeta(n+1,z)$ where $\Gamma$ is the gamma function and $\zeta$ the two argument Riemann zeta function. Assuming that this identity holds for the complex numbers, it may explain why you can't find any explicit code.

Looking at what the python scipy library does for its special functions, the polygamma is found by returning the digamma if the zeroth derivative is requested, otherwise return $(-1)^{n+1}\Gamma(n+1)\zeta(n+1,z)$ where $\Gamma$ is the gamma function and $\zeta$ the two argument Riemann zeta function. Assuming that this identity holds for the complex numbers, it may explain why you can't find any explicit code.

In fact to add a reference, this is also the method used in the gnu implementation of libstdc++, see here