imported>Sure Beae: /* Relations */ Fixed the improper algebra. The duplication formula introduced in the paper cannot be used like it was.

2025-01-30T19:59:12Z

Relations: Fixed the improper algebra. The duplication formula introduced in the paper cannot be used like it was.

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In mathematics, the '''generalized polygamma function''' or '''balanced negapolygamma function''' is a function introduced by Olivier Espinosa Aldunate and [[Victor Hugo Moll]].<ref>{{cite journal|url=http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf|first1=Olivier|last1=Espinosa|first2=Victor Hugo|last2=Moll |author-link2=Victor Hugo Moll |title=A Generalized polygamma function|journal=Integral Transforms and Special Functions|volume=15|issue=2|date=Apr 2004|pages=101–115|doi=10.1080/10652460310001600573 }}{{open access}}</ref>

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

==Definition==

The 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
:<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>\begin{align}
\psi(x) &= \psi(0,x)\\
\psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \\
\Gamma(x)&=\exp\left( \psi(-1,x)+\tfrac12 \ln 2\pi \right)\\
\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>K(z)=A \exp\left(\psi(-2,z)+\frac{z^2-z}{2}\right)</math>

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

==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>\begin{align}
\psi\left(-2,\tfrac14\right)&=\tfrac18\ln A+\frac{G}{4\pi} && \\
\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}\\
\psi(-2,1)&=-\ln A &\\
\psi(-3,1)&=\frac{-\zeta(3)}{8\pi^2}\\
\psi(-2,2)&=-\ln A-1 &\\
\psi(-3,2)&=\frac{-\zeta(3)}{8\pi^2}-\tfrac34 \\\end{align}</math>

==References==
<references />

{{DEFAULTSORT:Generalized Polygamma Function}}
[[Category:Gamma and related functions]]

Balanced polygamma function - Revision history

imported>Sure Beae: /* Relations */ Fixed the improper algebra. The duplication formula introduced in the paper cannot be used like it was.