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:

\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),

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

The function is balanced, in that it satisfies the conditions $f(0)=f(1)$ and $\int_0^1 f(x) dx = 0$ .

Relations

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

$\psi(x)=\psi(0,x)\,$
$\psi^{(n)}(x)=\psi(n,x)\,\,\,(n\in\mathbb{N})$
$\Gamma(x)=e^{\psi(-1,x)+\frac 12 \ln(2\pi)}\,\,\,$
$\zeta(z,q)=\frac{\Gamma (1-z) \left(2^{-z} \left(\psi \left(z-1,\frac{q}{2}+\frac{1}{2}\right)+\psi \left(z-1,\frac{q}{2}\right)\right)-\psi(z-1,q)\right)}{\ln(2)}$
$\zeta'(-1,x)=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12}$
$B_n(q) = -\frac{\Gamma (n+1) \left(2^{n-1} \left(\psi\left(-n,\frac{q}{2}+\frac{1}{2}\right)+\psi\left(-n,\frac{q}{2}\right)\right)-\psi(-n,q)\right)}{\ln (2)}$

where $B_n(q)$ are Bernoulli polynomials

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

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:

$\psi\left(-2,\frac14\right)=\frac18\ln(2\pi)+\frac98\ln A+\frac{G}{4\pi},$ where $A$ is the Glaisher constant and $G$ is the Catalan constant.
$\psi\left(-2, \frac12\right)=\frac14\ln\pi+\frac32\ln A+\frac5{24}\ln2$
$\psi(-2,1)=\frac12\ln(2\pi)$
$\psi(-2,2)=\ln(2\pi)-1$
$\psi\left(-3,\frac12\right)=\frac1{16}\ln(2\pi)+\frac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}$
$\psi(-3,1)=\frac14\ln(2\pi)+\ln A$
$\psi(-3,2)=\ln(2\pi)+2\ln A-\frac34$

References

↑ Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115

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