Balanced polygamma function: Difference between revisions

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In mathematics, the '''Generalizedgeneralized polygamma function''' or '''balanced negapolygamma function''' is a function, introduced by Olivier Espinosa Aldunate and [[Victor H.Hugo Moll]].<ref>Olivier{{cite Espinosa Victor H. Moll. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115 [journal|url=http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf]</ref>|first1=Olivier|last1=Espinosa|first2=Victor ItHugo|last2=Moll generalizes|author-link2=Victor theHugo [[PolygammaMoll function]]|title=A toGeneralized negativepolygamma andfunction|journal=Integral fractionalTransforms order,and butSpecial remainsFunctions|volume=15|issue=2|date=Apr equal2004|pages=101–115|doi=10.1080/10652460310001600573 to}}{{open it for integer positive orders.access}}</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:

==Definition==

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

ItThe generalized polygamma function is defined as follows:

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

where {{math|''ψ''(''z'')}} is the [[polygamma function]] and {{math|''ζ''(''z'',''q'')}}, is the [[Hurwitz zeta function]].

The function is balanced, in that it satisfies the conditions

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

:<math>f(0)=f(1) \quad \text{and} \quad \int_0^1 f(x)\, dx = 0</math>.

==Relations==

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

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

:<math>\begin{align}

\psi(x) &= \psi(0,x)\\

\psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \\

* <math>\Gamma(x)&=e^{\exp\left( \psi(-1,x)+\frac 12tfrac12 \ln( 2\pi \right)}\,\,\,</math>

\zeta(z, q)&=\frac{(-1)^z}{\Gamma(z)} \psi(z - 1, q)\\

\zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12} \\

\end{align}</math>

* :<math>\zetaK(z,q)=\frac{\Gamma (1-z)A \left(2^{-z} exp\left(\psi \left(z-12,\frac{q}{2}z)+\frac{1}{z^2}\right)+\psi \left(-z-1,\frac{q}{2}\right)\right)-\psi(z-1,q)\right)}{\ln(2)}</math>

where {{math|''K''(''z'')}} is the [[K-function|{{mvar|K}}-function]] and {{mvar|A}} is the [[Glaisher constant]].

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

==Special values==

The balanced polygamma function can be expressed in a closed form at certain points (where {{mvar|A}} is the [[Glaisher constant]] and {{mvar|G}} is the [[Catalan constant]]):

* <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>

:<math>\begin{align}

\psi\left(-2,\tfrac14\right)&=\tfrac18\ln A+\frac{G}{4\pi} && \\

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

\psi\left(-2,\tfrac12\right)&=\tfrac12\ln A-\tfrac{1}{24}\ln 2 & \\

\psi\left(-3,\tfrac12\right)&=\frac{3\zeta(3)}{32\pi^2}\\

* <math>K(z)=\exp\left(\frac{z-z^2}{2}-\psi(-2,z1)&=-\ln A &\\right)</math>

\psi(-3,1)&=\frac{-\zeta(3)}{8\pi^2}\\

\psi(-2,2)&=-\ln A-1 &\\

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

\psi(-3,2)&=\frac{-\zeta(3)}{8\pi^2}-\tfrac34 \\\end{align}</math>

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

==References==

<references />

{{DEFAULTSORT:Generalized Polygamma Function}}

[[Category:SpecialGamma and related functions]]