Balanced polygamma function

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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.^[1]

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

Definition#edit

The generalized polygamma function is defined as follows:

${\displaystyle \psi (z,q)={\frac {\zeta '(z+1,q)+{\bigl (}\psi (-z)+\gamma {\bigr )}\zeta (z+1,q)}{\Gamma (-z)$

or alternatively,

${\displaystyle \psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)\right),}$

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

${\displaystyle f(0)=f(1)\quad {\text{and\quad \int _{0}^{1}f(x)\,dx=0}$ .

Relations#edit

Several special functions can be expressed in terms of generalized polygamma function.

${\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2\ln 2\pi \right)\\\zeta (z,q)&={\frac {(-1)^{z{\Gam$

${\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2\right)}$

where K(z) is the K-function and A is the Glaisher constant.

Special values#edit

The balanced polygamma function can be expressed in a closed form at certain points (where A is the Glaisher constant and G is the Catalan constant):

${\displaystyle {\begin{aligned}\psi \left(-2,{\tfrac {1}{4\right)&={\tfrac {1}{8\ln A+{\frac {G}{4\pi &&\\\psi \left(-2,{\tfrac {1}{2\right)&={\tfrac {1}{2\ln A-{\tfra$

References#edit

^ Espinosa, Olivier; Moll, Victor Hugo (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115. doi:10.1080/10652460310001600573.

Categories Gamma and related functions

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