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→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
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)+\
or alternatively,
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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|''ψ''(''z'')}} is the [[
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{
\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>
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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
\psi\left(-2,\tfrac12\right)&=\
\psi\left(-3,\tfrac12\right)&=
\psi(-2,1)&=
\psi(-3,1)&=\
\psi(-2,2)&=-\ln
\psi(-3,2)&=\
==References==
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