PolyGamma Functions of Negative Order

Liouville’s fractional integration is used to define polygamma functions $\psi^{(n)}(z)$ for negative integer $n$. It’s shown that such $\psi^{(n)}(z)$ can be represented in a closed form by means of the first derivatives of the Hurwitz Zeta function. Relations to the Barnes G-function and generalized Glaisher’s constants are also discussed.