Balanced polygamma function: Difference between revisions

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In mathematics, the '''generalized polygamma function''' or '''balanced negapolygamma function''' is a function introduced by Olivier Espinosa Aldunate 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. The function is balanced, that is satisfies the conditions <math>f(0)=f(1)</math> and <math>\int_0^1 f(x) dx = 0</math>.

It generalizes the [[polygamma function]] to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

==Definition==

ItThe generalized polygamma function is defined as follows:

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

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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>

:where <math>\psi(z)</math> is the [[Polygamma function]] and <math>\zeta(z,q),</math> is the [[Hurwitz zeta function]].

The function is balanced, in that it satisfies the conditions <math>f(0)=f(1)</math> and <math>\int_0^1 f(x) dx = 0</math>.

==Relations==

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

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* <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 zeta function]]

* <math>\zeta'(-1,x)=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12}</math>

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

:where ''K''(''z'') is [[K-function]] and A is [[Glaisher constant]].

==References==