In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.
[1] It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
The generalized polygamma function is defined as follows: or alternatively, where ψ(z) is the polygamma function and ζ(z,q), is the Hurwitz zeta function.
where K(z) is the K-function and A is the Glaisher constant.
The balanced polygamma function can be expressed in a closed form at certain points (where A is the Glaisher constant and G is the Catalan constant):