Balanced polygamma function

Definition

The generalized polygamma function is defined as follows:

: \psi(z,q)=\frac{\zeta'(z+1,q)+\bigl(\psi(-z)+\gamma \bigr) \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 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 :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.

:\begin{align} \psi(x) &= \psi(0,x)\ \psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \ \Gamma(x)&=\exp\left( \psi(-1,x)+\tfrac12 \ln 2\pi \right)\ \zeta(z, q)&=\frac{(-1)^z}{\Gamma(z)} \psi(z - 1, q)\ \zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12} \ \end{align}

: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): :\begin{align} \psi\left(-2,\tfrac14\right)&=\tfrac18\ln A+\frac{G}{4\pi} && \ \psi\left(-2,\tfrac12\right)&=\tfrac12\ln A-\tfrac{1}{24}\ln 2 & \ \psi\left(-3,\tfrac12\right)&=\frac{3\zeta(3)}{32\pi^2}\ \psi(-2,1)&=-\ln A &\ \psi(-3,1)&=\frac{-\zeta(3)}{8\pi^2}\ \psi(-2,2)&=-\ln A-1 &\ \psi(-3,2)&=\frac{-\zeta(3)}{8\pi^2}-\tfrac34 \\end{align}

References

References

1. (Apr 2004). "A Generalized polygamma function". Integral Transforms and Special Functions.