Balanced polygamma function

In mathematics, the generalized polygamma function or balanced negapolygamma 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. The function is balanced, that is satisfies the conditions $f(0)=f(1)$ and $\int_0^1 f(x) dx = 0$.

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.

• $\psi (x)=\psi (0,x)\,$

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

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

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

where $\zeta (z,q),$ is the 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

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

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

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