Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Makoto Kawashima, Anthony Poëls

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.

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

Abstract

In this article, we present a new linear independence criterion for values of the -adic polygamma functions defined by J.~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 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 -adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.

Paper Structure

This paper contains 25 sections, 289 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. The $p$--adic Hurwitz zeta function
  4. Padé approximation theory
  5. Modified formal transforms
  6. Formal $f$--integration transform
  7. Notation and definitions
  8. Properties of the operators $D$ and $D^*$
  9. Properties of the transform $\varphi_f$
  10. Mellin transform and $p$--adic polygamma functions
  11. Properties of the difference operator
  12. Construction of Padé approximants
  13. Kernel of the formal integration maps
  14. Study of the kernels
  15. Proof of Theorem \ref{['thm: non-singular matrix for non-vanishing of the det']}
  16. ...and 10 more sections

Figures (2)

  • Figure 1: Comparison between our condition and that of P. Bel
  • Figure 2: Expected characteristic polynomials for small values of $M$

Theorems & Definitions (46)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 36 more