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.Hugo Moll]].<ref>{{cite journal|url=http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf|first1=Olivier|last1=Espinosa|first2=Victor H.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|pagepages=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.

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The generalized polygamma function is defined as follows:

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

or alternatively,

: <math>\psi(z,q)&nbsp;<s>^{<pre><blockquote><ref name="{{#tag:ref|<ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref><ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref></ref>|group="nb"|name=""}}" /></blockquote></pre>}</s>=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 [[Polygammapolygamma function]] and {{math|''ζ''(''z'',''q'')}}, is the [[Hurwitz zeta function]].

The function is balanced, in that it satisfies the conditions

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\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{\Gamma (-1-z)}{\ln 2} \left(2^{-z} {\psi \leftGamma(z-1,\frac{q+1}{2}\right)+2^{-z} \psi \left(z - 1,\frac{ q}{2}\right)-\psi(z-1,q)\right)\\

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

B_n(q) &= -\frac{\Gamma (n+1)}{\ln 2} \left(2^{n-1} \psi\left(-n,\frac{q+1}{2}\right)+2^{n-1} \psi\left(-n,\frac{q}{2}\right)-\psi(-n,q)\right)

\end{align}</math>

where {{math|''B_n''(''q'')}} are [[Bernoulli polynomials]]

:<math>K(z)=A \exp\left(\psi(-2,z)+\frac{z^2-z}{2}\right)</math>

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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 2\pi+\tfrac98\ln A+\frac{G}{4\pi} && \\

\psi\left(-2,\tfrac12\right)&=\tfrac14\ln\pi+\tfrac32tfrac12\ln A+-\tfrac5tfrac{1}{24}\ln2ln 2 & \\

\psi\left(-3,\tfrac12\right)&=\tfrac1{16}\ln 2\pi+\tfrac12\ln A+\frac{73\zeta(3)}{32\pi^2}\\

\psi(-2,1)&=\tfrac12-\ln 2\piA &\\

\psi(-3,1)&=\tfrac14frac{-\ln 2zeta(3)}{8\pi+\ln A^2}\\

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

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

==References==