Balanced polygamma function

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.

Definition

edit

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

 .

Relations

edit

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

 

where Bn(q) are the 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):

 

References

edit
  1. ^ Espinosa, Olivier; Moll, Victor Hugo (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115. doi:10.1080/10652460310001600573.