Balanced polygamma function

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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.

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 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 H. (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115. open access publication – free to read