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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 802791, 7 pages
http://dx.doi.org/10.1155/2013/802791
Research Article

Algorithms for Some Euler-Type Identities for Multiple Zeta Values

Department of Mathematics, Central South University, Changsha, Hunan 410083, China

Received 21 December 2012; Accepted 11 January 2013

Academic Editor: C. Conca

Copyright © 2013 Shifeng Ding and Weijun Liu. 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.

Abstract

Multiple zeta values are the numbers defined by the convergent series , where , , , are positive integers with . For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is . The well-known result was extended to and by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers and then gave a direct formula for for arbitrary . In this paper we apply a technique introduced by Granville to present an algorithm to calculate and prove that the direct formula can also be deduced from Eisenstein's double product.