Balanced polygamma function

Generalized polygamma function is a function, introduced by Olivier Espinosa and Victor H. Moll It generalizes the Polygamma function to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

$\psi (z,q)={\frac {\zeta '(z+1,q)+(\psi (-z)+\gamma )\zeta (z+1,q)}{\Gamma (-z)}}$

or alternatively,

$\psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z}}\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)}}\right)$