Balanced polygamma function

Generalized polygamma function is a function, introduced by Olivier Espinosa and Victor H. Moll^[1] 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,

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

References

^ Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115 [1]

[1] Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115 [1]

[1]