Balanced polygamma function
Open-knowledge reference entry
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.
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:
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{\displaystyle \psi (z,q)={\frac {\zeta '(z+1,q)+{\bigl (}\psi (-z)+\gamma {\bigr )}\zeta (z+1,q)}{\Gamma (-z)}}}
or alternatively,
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{\displaystyle \psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z}}\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)}}\right),}
where ψ(z) is the polygamma function and ζ(z,q), is the Hurwitz zeta function.
The function is balanced, in that it satisfies the conditions
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{\displaystyle f(0)=f(1)\quad {\text{and}}\quad \int _{0}^{1}f(x)\,dx=0}
.
Relations
Several special functions can be expressed in terms of generalized polygamma function.
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{\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\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}}+{\frac {1}{12}}\\\end{aligned}}}
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{\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2}}\right)}
where K(z) is the K-function and A is the Glaisher constant.
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