On a Reciprocity Law for Finite Multiple Zeta Values

Abstract

It was shown by Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) 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 Kirschenhofer and Prodinger (1998) and Kuba et al. (2008) 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 combinatorial proof of the shuffle identity based on partial fraction decomposition. We also present an extension of the reciprocity relation to weighted sums.