International Journal of Combinatorics
Volume 2010 (2010), Article ID 153621, 13 pages
Research Article

On a Reciprocity Law for Finite Multiple Zeta Values

1Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr, 8-10/104, 1040 Wien, Austria
2Department of Mathematics, University of Stellenbosch, 7602 Stellenbosch, South Africa

Received 11 October 2009; Accepted 14 January 2010

Academic Editor: Alois Panholzer

Copyright © 2010 Markus Kuba and Helmut Prodinger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.