A linear independence criterion for $p$-adic Hurwitz zeta values at odd integers

In this article, a new criterion is established to obtain the linear independence of some values of the $p$-adic Hurwitz zeta function at odd integers $s\geq 3$. We generalize a theorem of F.~Beukers, who gave, among others, an alternative proof---using Padé approximation theory---of a result origin