On a reciprocity law for finite multiple zeta values

TL;DR

This paper generalizes a reciprocity law to finite multiple zeta values, providing new identities and elementary proofs, which could impact the analysis of algorithms and number theory.

Contribution

It introduces a generalized reciprocity relation for finite multiple zeta values and offers a simple proof of the shuffle identity using partial fractions.

Findings

Generalized reciprocity relation for finite multiple zeta values.

Elementary proof of the shuffle identity via partial fractions.

Extension of reciprocity to weighted sums.

Abstract

It was shown in that harmonic numbers satisfy certain reciprocity relations, which are in particular useful for the analysis of the quickselect algorithm. The aim of this work is to show that a reciprocity relation from \cite{KirProd98,ProSchnKu} can be generalized to finite variants of multiple zeta values, involving a finite variant of the shuffle identity for multiple zeta values. We present the generalized reciprocity relation and furthermore a simple elementary proof of the shuffle identity using only partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.