2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/173554We study the properties of a function $ψ(z, q)$ (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 $ψ^{(m)}(q)$ for $z$ a non-negative integer $m$, and to the balanced negapolygamma function $ψ^{(-m)}(q)$ for $z$ a negative integer $-m$.14 pagesClassical Analysis and ODEsMathematical Physics33B99, 11M35text