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$p$-adic multiple zeta values of integer indices

Ku-Yu Fan

Abstract

This paper concerns the $p$-adic multiple zeta values of integer indices that may contain zero or negative components. We introduce the admissibility and regularizability conditions for integer indices. We define the $p$-adic multiple zeta values associated with admissible integer indices to be finite rational linear combinations of $p$-adic multiple zeta values associated with admissible positive integer indices. We prove that the double shuffle relations, that is, the shuffle and stuffle product formulas, both hold for the values.

$p$-adic multiple zeta values of integer indices

Abstract

This paper concerns the -adic multiple zeta values of integer indices that may contain zero or negative components. We introduce the admissibility and regularizability conditions for integer indices. We define the -adic multiple zeta values associated with admissible integer indices to be finite rational linear combinations of -adic multiple zeta values associated with admissible positive integer indices. We prove that the double shuffle relations, that is, the shuffle and stuffle product formulas, both hold for the values.
Paper Structure (6 sections, 17 theorems, 92 equations)

This paper contains 6 sections, 17 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Integer indices
  3. MPLs of integer indices
  4. Shuffle product for integer indices
  5. Stuffle product for integer indices
  6. Applications

Key Result

Lemma 3.4

Let $k$ be a non-negative integer.

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 59 more