`International Journal of CombinatoricsVolume 2010 (2010), Article ID 153621, 13 pageshttp://dx.doi.org/10.1155/2010/153621`
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

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.

1. P. Kirschenhofer and H. Prodinger, “Comparisons in Hoare's find algorithm,” Combinatorics, Probability and Computing, vol. 7, no. 1, pp. 111–120, 1998.
2. C. A. R. Hoare, “Algorithm 64: quicksort,” Communications of the ACM, vol. 4, no. 7, pp. 321–322, 1961.
3. M. Kuba, H. Prodinger, and C. Schneider, “Generalized reciprocity laws for sums of harmonic numbers,” Integers: The Electronic Journal of Combinatorial Number Theory, vol. 8, no. A17, pp. 1–20, 2008.
4. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, “Combinatorial aspects of multiple zeta values,” Electronic Journal of Combinatorics, vol. 5, p. R38, 1998.
5. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lisonek, “Special values of multiple polylogarithms,” Transactions of the American Mathematical Society, vol. 353, no. 3, pp. 907–941, 2001.
6. D. Bowman and D. M. Bradley, “Multiple polylogarithms: a brief survey,” in Q-Series with Applications to Combinatorics, Number Theory, and Physics, vol. 291 of Contemporary Mathematics, pp. 71–92, 2001.
7. D. Bowman and D. M. Bradley, “The algebra and combinatorics of shuffles and multiple zeta values,” Journal of Combinatorial Theory A, vol. 97, no. 1, pp. 43–61, 2002.
8. D. Bowman, D. M. Bradley, and J. H. Ryoo, “Some multi-set inclusions associated with shuffle convolutions and multiple zeta values,” European Journal of Combinatorics, vol. 24, no. 1, pp. 121–127, 2003.
9. D. Zagier, “Values of zeta functions and their applications,” in First European Congress of Mathematics Vol. II, pp. 497–512, Birkhäuser, Boston, Mass, USA, 1994.
10. J. Blümlein and S. Kurth, “Harmonic sums and Mellin transforms up to two-loop order,” Physical Review D, vol. 60, Article ID 014018, 1999.
11. J. A. M. Vermaseren, “Harmonic sums, Mellin transforms and integrals,” International Journal of Modern Physics A, vol. 14, no. 13, pp. 2037–2076, 1999.
12. S. Moch, P. Uwer, and S. Weinzierl, “Nested sums, expansion of transcendental functions, and multiscale multiloop integrals,” Journal of Mathematical Physics, vol. 43, no. 6, pp. 3363–3386, 2002.
13. M. E. Hoffman, “Quasi-shuffle products,” Journal of Algebraic Combinatorics, vol. 11, no. 1, pp. 49–68, 2000.
14. T. H. Koornwinder and M. Schlosser, “On an identity by Chaundy and Bullard,” Indagationes Mathematicae. New Series, vol. 19, no. 2, pp. 239–261, 2008.
15. N. Nielsen, Handbuch der Theorie der Gamma Funktion, Chelsea, New York, NY, USA, 1965.
16. N. M. Hoang and M. Petitot, “Lyndon words, polylogarithms and the Riemann $\zeta$ function,” Discrete Mathematics, vol. 217, no. 1–3, pp. 273–292, 2000.
17. N. M. Hoang, M. Petitot, and J. Van Der Hoeven, “Shuffle algebra and polylogarithms,” in Proceedings of the 10th Conference on Formal Power Series and Algebraic Combinatorics (FPSAC '98), p. 12, Toronto, Canada, June 1998.