Balanced polygamma function

In mathematics, the generalized polygamma function or balanced negapolygamma 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

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

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

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

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