A generalized polygamma function

Abstract

We study the properties of a function $\psi (z,q)$ (the generalized polygamma function), intimately connected with the Hurwitz zeta function and defined for complex values of the variables $z$ and $q$, which is entire in the variable $z$ and reduces to the usual polygamma function ${\psi }^{(m)}(q)$ for $z$ a non-negative integer $m$, and to the balanced negapolygamma function ${\psi }^{(-m)}(q)$ for $z$ a negative integer $-m$.

14 pages