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 is the Polygamma function and is the Hurwitz zeta function.

The function is balanced, in that it satisfies the conditions and .

Relations[edit]

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

where are Bernoulli polynomials

where K(z) is 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 is the Glaisher constant and is the Catalan constant.

References[edit]

  1. ^ Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115