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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.
==Definition==
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: <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>.
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* <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>
* <math>\zeta'(-1,x)=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12}</math>
* <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>
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The balanced polygamma function can be expressed in a closed form at certain points:
* <math>\psi\left(-2,\frac14\right)=\frac18\ln(2\pi)+\frac98\ln A+\frac{G}{4\pi},</math> where <math>A</math> is the [[Glaisher constant]] and <math>G</math> is the [[Catalan constant]].
* <math>\psi\left(-2, \frac12\right)=\frac14\ln\pi+\frac32\ln A+\frac5{24}\ln2</math>
* <math>\psi(-2,1)=\frac12\ln(2\pi)</math>
* <math>\psi(-2,2)=\ln(2\pi)-1</math>
* <math>\psi\left(-3,\frac12\right)=\frac1{16}\ln(2\pi)+\frac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}</math>
* <math>\psi(-3,1)=\frac14\ln(2\pi)+\ln A</math>
* <math>\psi(-3,2)=\ln(2\pi)+2\ln A-\frac34</math>
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