Year: 2024
Type: preprint
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 $ζ_p(s,x)$ 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 Padé-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.
Source: arXiv (Cornell University)
Authors: Kawashima, Makoto, Poëls, Anthony
Institutions: Institut Camille Jordan, Meiji Gakuin University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403345474