A generalized polygamma function

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 $-m$.