  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 $-m$.
