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

Abstract

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 originally due to F.~Calegari on the irrationality of the $p$-adic Riemann zeta value $ζ_p(3)$ for $p = 2, 3$. To prove our main theorem, we work in a general setting, where we construct and study new Padé approximants of Laurent series satisfying the functional equation $f(z) = f(1-z)$. This complements our previous work on the linear independence of $p$-adic polygamma values.