Balanced polygamma function

In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor H. Moll.^[1]

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:

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

or alternatively,

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

where $\psi (z)$ is the Polygamma function and $\zeta (z,q),$ is the Hurwitz zeta function.

The function is balanced, in that it satisfies the conditions $f(0)=f(1)$ and $\int _{0}^{1}f(x)dx=0$ .

Relations

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

$\psi (x)=\psi (0,x)\,$

$\psi ^{(n)}(x)=\psi (n,x)\,\,\,(n\in \mathbb {N} )$

$\Gamma (x)=e^{\psi (-1,x)+{\frac {1}{2}}\ln(2\pi )}\,\,\,$

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

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

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

where $B_{n}(q)$ are Bernoulli polynomials

$K(z)=Ae^{\psi (-2,z)+{\frac {z^{2}-z}{2}}}$

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

Special values

The balanced polygamma function can be expressed in a closed form at certain points:

$\psi ^{(-2)}\left({\frac {1}{4}}\right)={\frac {1}{8}}\ln(2\pi )+{\frac {9}{8}}\ln A+{\frac {G}{4\pi }},$ where $A$ is the Glaisher constant and $G$ is the Catalan constant.