Balanced polygamma function

Generalized polygamma function is a function, introduced by Olivier Espinosa and Victor H. Moll It generalizes the Polygamma function to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

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

or alternatively,

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

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

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

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

where $\zeta (z,q),$ is Hurwitz Zeta function

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

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

• $K(z)={\frac {e^{{\frac {z-z^{2}}{2}}-\psi (-2,z)}}{A}}$

where K(z) is K-function ana A is Glaisher constant, which itself can be expressed in terms of generalized polygamma function:

• $A={\frac {\sqrt[{36}]{128{\pi }^{30}}}{\pi }}e^{{\frac {1}{3}}+{\frac {2}{3}}\psi (-1,{\frac {1}{2}})-{\frac {1}{3}}\ln(2\pi )}$