On the linear independence of pp‐adic polygamma values

In this article, we present a new linear independence criterion for values of the ‐adic polygamma functions defined by Diamond. As an application, we obtain the linear independence of some families of values of the ‐adic Hurwitz zeta function at distinct shifts . This improves and extends a previous result due to Bel (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) IX (2010), 189–227), as well as irrationality results established by Beukers (Acta Math. Sin. 24 (2008), 663–686). Our proof is based on a novel and explicit construction of Padé‐type approximants of the second kind of Diamond's ‐adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.