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:

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

or alternatively,

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

where ${\displaystyle \psi (z)}$  is the Polygamma function and ${\displaystyle \zeta (z,q),}$  is the Hurwitz zeta function.

The function is balanced, in that it satisfies the conditions ${\displaystyle f(0)=f(1)}$  and ${\displaystyle \int _{0}^{1}f(x)dx=0}$ .

Relations

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

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

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

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

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

• ${\displaystyle \zeta '(-1,x)=\psi (-2,x)+{\frac {x^{2}}{2}}-{\frac {x}{2}}+{\frac {1}{12}}}$ 

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

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 the Glaisher constant.

Special values

The balanced polygamma function can be expressed in a closed form at certain points:

• ${\displaystyle \psi \left(-2,{\frac {1}{4}}\right)={\frac {1}{8}}\ln(2\pi )+{\frac {9}{8}}\ln A+{\frac {G}{4\pi }},}$  where ${\displaystyle A}$  is the Glaisher constant and ${\displaystyle G}$  is the Catalan constant.

• ${\displaystyle \psi \left(-2,{\frac {1}{2}}\right)={\frac {1}{4}}\ln \pi +{\frac {3}{2}}\ln A+{\frac {5}{24}}\ln 2}$ 

• ${\displaystyle \psi (-2,1)={\frac {1}{2}}\ln(2\pi )}$ 

• ${\displaystyle \psi (-2,2)=\ln(2\pi )-1}$ 

• ${\displaystyle \psi \left(-3,{\frac {1}{2}}\right)={\frac {1}{16}}\ln(2\pi )+{\frac {1}{2}}\ln A+{\frac {7\,\zeta (3)}{32\,\pi ^{2}}}}$ 

• ${\displaystyle \psi (-3,1)={\frac {1}{4}}\ln(2\pi )+\ln A}$ 

• ${\displaystyle \psi (-3,2)=\ln(2\pi )+2\ln A-{\frac {3}{4}}}$ 

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