In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.[1]
It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
The generalized polygamma function is defined as follows:
| \psi(z,q)= | \zeta'(z+1,q)+l(\psi(-z)+\gammar)\zeta(z+1,q) |
| \Gamma(-z) |
\psi(z,q)=e-
| \partial | |
| \partialz |
\left(e\gamma
| \zeta(z+1,q) | |
| \Gamma(-z) |
\right),
where is the polygamma function and, is the Hurwitz zeta function.
The function is balanced, in that it satisfies the conditions
f(0)=f(1) and
| 1 | |
| \int | |
| 0 |
f(x)dx=0
Several special functions can be expressed in terms of generalized polygamma function.
\begin{align} \psi(x)&=\psi(0,x)\\ \psi(n)(x)&=\psi(n,x) n\inN\\ \Gamma(x)&=\exp\left(\psi(-1,x)+\tfrac12ln2\pi\right)\\ \zeta(z,q)&=
| \Gamma(1-z) | |
| ln2 |
\left(2-z\psi\left(z-1,
| q+1 | |
| 2 |
\right)+2-z\psi\left(z-1,
| q | |
| 2 |
\right)-\psi(z-1,q)\right)\\ \zeta'(-1,x)&=\psi(-2,x)+
| x2 | |
| 2 |
-
| x | |
| 2 |
+
| 1{12} | |
| \\ B |
n(q)&=-
| \Gamma(n+1) | |
| ln2 |
\left(2n-1\psi\left(-n,
| q+1 | |
| 2 |
\right)+2n-1\psi\left(-n,
| q | |
| 2 |
\right)-\psi(-n,q)\right) \end{align}
where are the Bernoulli polynomials
K(z)=A\exp\left(\psi(-2,z)+
| z2-z | |
| 2 |
\right)
where is the -function and is the Glaisher constant.
The balanced polygamma function can be expressed in a closed form at certain points (where is the Glaisher constant and is the Catalan constant):
\begin{align} \psi\left(-2,\tfrac14\right)&=\tfrac18ln2\pi+\tfrac98lnA+
| G | |
| 4\pi |
&&\\ \psi\left(-2,\tfrac12\right)&=\tfrac14ln\pi+\tfrac32lnA+\tfrac5{24}ln2&\\ \psi\left(-3,\tfrac12\right)&=\tfrac1{16}ln2\pi+\tfrac12lnA+
| 7\zeta(3) | |
| 32\pi2 |
\\ \psi(-2,1)&=\tfrac12ln2\pi&\\ \psi(-3,1)&=\tfrac14ln2\pi+lnA\\ \psi(-2,2)&=ln2\pi-1&\\ \psi(-3,2)&=ln2\pi+2lnA-\tfrac34\\\end{align}