In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa and Victor H. Moll. It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

or alternatively,

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

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where is the Hurwitz zeta function

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where are Bernoulli polynomials

Bernoulli polynomials

In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator...

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where K(z) is K-function

K-function

In mathematics, the K-function, typically denoted K, is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function.Formally, the K-function is defined as...

 and A is Glaisher constant, which itself can be expressed in terms of generalized polygamma function:

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