Balanced polygamma function: Difference between revisions

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In mathematics, the '''Generalizedgeneralized polygamma function''' is a function, introduced by Olivier Espinosa and 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 [[Polygammapolygamma function]] to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

: <math>\psi(z,q)=\frac{\zeta'(z+1,q)+(\psi(-z)+\gamma ) \zeta (z+1,q)}{\Gamma (-z)} \, </math>

<math>\psi(z,q)=\frac{\zeta'(z+1,q)+(\psi(-z)+\gamma ) \zeta (z+1,q)}{\Gamma (-z)}</math>

or alternatively,

: <math>\psi(z,q)=e^{- \gamma z}\frac{\partial}{\partial z}\left(e^{\gamma z}\frac{\zeta(z+1,q)}{\Gamma(-z)}\right)</math>

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

* <math>\psi(x)=\psi(0,x)\,</math>

* <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>

:where <math>\zeta(z,q),</math> is the [[Hurwitz Zetazeta function]]

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

:where <math>B_n(q)</math> are [[Bernoulli polynomials]]

* <math>K(z)=\frac {e^{\frac{z-z^2}{2}-\psi(-2,z)}}A</math>

:where K(z) is [[K-function]] ana A is [[Glaisher constant]], which itself can be expressed in terms of generalized polygamma function:

*<math>A =\frac{\sqrt[36]{128{\pi}^{30}}}{\pi}e^{\frac{1}{3}+\frac{2}{3}\psi(-1,\frac 12)-\frac 13\ln(2\pi)}</math>