# 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:

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

or alternatively,

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

where ψ(z) is the Polygamma function and ζ(z,q), is the Hurwitz zeta function.

The function is balanced, in that it satisfies the conditions

${\displaystyle f(0)=f(1)\quad {\text{and}}\quad \int _{0}^{1}f(x)\,dx=0}$.

## Relations

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

{\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\[8px]\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\[8px]\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\ln 2\pi \right)\\[8px]\zeta (z,q)&={\frac {\Gamma (1-z)}{\ln 2}}\left(2^{-z}\psi \left(z-1,{\frac {q+1}{2}}\right)+2^{-z}\psi \left(z-1,{\frac {q}{2}}\right)-\psi (z-1,q)\right)\\[8px]\zeta '(-1,x)&=\psi (-2,x)+{\frac {x^{2}}{2}}-{\frac {x}{2}}+{\frac {1}{12}}\\[8px]B_{n}(q)&=-{\frac {\Gamma (n+1)}{\ln 2}}\left(2^{n-1}\psi \left(-n,{\frac {q+1}{2}}\right)+2^{n-1}\psi \left(-n,{\frac {q}{2}}\right)-\psi (-n,q)\right)\end{aligned}}}

where Bn(q) are Bernoulli polynomials

${\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2}}\right)}$

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

## Special values

The balanced polygamma function can be expressed in a closed form at certain points (where A is the Glaisher constant and G is the Catalan constant):

{\displaystyle {\begin{aligned}\psi \left(-2,{\tfrac {1}{4}}\right)&={\tfrac {1}{8}}\ln 2\pi +{\tfrac {9}{8}}\ln A+{\frac {G}{4\pi }}&&\\[8px]\psi \left(-2,{\tfrac {1}{2}}\right)&={\tfrac {1}{4}}\ln \pi +{\tfrac {3}{2}}\ln A+{\tfrac {5}{24}}\ln 2&\psi \left(-3,{\tfrac {1}{2}}\right)&={\tfrac {1}{16}}\ln 2\pi +{\tfrac {1}{2}}\ln A+{\frac {7\zeta (3)}{32\pi ^{2}}}\\[8px]\psi (-2,1)&={\tfrac {1}{2}}\ln 2\pi &\psi (-3,1)&={\tfrac {1}{4}}\ln 2\pi +\ln A\\[8px]\psi (-2,2)&=\ln 2\pi -1&\psi (-3,2)&=\ln 2\pi +2\ln A-{\tfrac {3}{4}}\end{aligned}}}

## References

1. ^ Espinosa, Olivier; Moll, Victor H. (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115.