Balanced polygamma function

Generalized polygamma function is a function, introduced by Olivier Espinosa and Victor H. Moll^[1] It generalizes the Polygamma function to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

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

or alternatively,

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

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

• ${\displaystyle \psi (x)=\psi (0,x)\,}$

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

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

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

where ${\displaystyle \zeta (z,q),}$ is the Hurwitz Zeta function

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

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

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

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

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

References

1. Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115