Balanced polygamma function: Difference between revisions

Revision as of 23:32, 15 June 2010

Revision as of 22:12, 26 December 2010

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In mathematics, the '''generalized polygamma function''' or '''balanced negapolygamma function''' is a function introduced by Olivier Espinosa and [[Victor Moll|Victor H. Moll]].<ref>[http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115]</ref>. It generalizes the [[polygamma function]] to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

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* <math>\psi^{(n)}(x)=\psi(n,x)\,\,\,(n\in\mathbb{N})</math>

* <math>\Gamma(x)=e^{\psi(-1,x)+\frac 12 \ln(2\pi)}\,\,\,</math>

* <math>\zeta(z,q)=\frac{\Gamma (1-z) \left(2^{-z} \left(\psi \left(z-1,\frac{q}{2}+\frac{1}{2}\right)+\psi \left(z-1,\frac{q}{2}\right)\right)-\psi(z-1,q)\right)}{\ln(2)}</math>

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<references />

{{DEFAULTSORT:Generalized Polygamma Function}}

[[Category:Special functions]]

[[Category:Gamma and related functions]]