On the linear independence of $p$-adic polygamma values

TL;DR

This paper introduces a new criterion for linear independence of $p$-adic polygamma values, leading to results on the independence and irrationality of certain $p$-adic zeta function values, using novel Padé approximants.

Contribution

It develops a new linear independence criterion for $p$-adic polygamma functions and constructs explicit Padé approximants to prove independence and irrationality results.

Findings

Proved linear independence of certain $p$-adic Hurwitz zeta values.

Extended previous results by Bel and Beukers.

Constructed explicit Padé approximants using orthogonal polynomial techniques.

Abstract

In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $\zeta_p(s,x)$ at distinct shifts $x$. This improves and extends a previous result due to P.~Bel [5], as well as irrationality results established by F.~Beukers [7]. Our proof is based on a novel and explicit construction of Pad\'{e}-type approximants of the second kind of Diamond's $p$-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.