Balanced polygamma function

From Wikipedia, the free encyclopedia
  (Redirected from Generalized polygamma function)
Jump to: navigation, search

In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor H. 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.

The function is balanced, in that it satisfies the conditions



Several special functions can be expressed in terms of generalized polygamma function.

where Bn(q) are Bernoulli polynomials

where K(z) is the K-function and A is the Glaisher constant.

Special values[edit]

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):


  1. ^ Espinosa, Olivier; Moll, Victor H. (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115. open access publication – free to read