$p$-adic multiple zeta values of integer indices

TL;DR

This paper extends the theory of $p$-adic multiple zeta values to include integer indices with zero or negative components, establishing foundational properties and relations such as double shuffle formulas.

Contribution

It introduces admissibility and regularizability conditions for integer indices and proves the double shuffle relations for these extended $p$-adic multiple zeta values.

Findings

Double shuffle relations hold for extended $p$-adic multiple zeta values.

Defined $p$-adic multiple zeta values for admissible integer indices.

Established finiteness and rational linear combination properties.

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.