Algorithms for Some Euler-Type Identities for Multiple Zeta Values. (4th February 2013)

• Record Type: Journal Article
• Authors: Ding, Shifeng; Liu, Weijun
• Other Names: Conca C. Academic Editor.
• Is Part Of: Journal of applied mathematics. Volume 2013(2013)
• Publication Date: 2013-02-04
• Subjects: Mathematics -- Periodicals; 519.05
• DOI: 10.1155/2013/802791
• Languages: English
• ISSNs: 1110-757X

Abstract

Abstract : Multiple zeta values are the numbers defined by the convergent series ζ ( s 1, s 2, …, s k ) = ∑ n 1 > n 2 > ⋯ > n k > 0 ( 1 / n 1 s 1 n 2 s 2 ⋯ n k s k ), where s 1, s 2, …, s k are positive integers with s 1 > 1 . For k ≤ n, let E ( 2 n, k ) be the sum of all multiple zeta values with even arguments whose weight is 2 n and whose depth is k . The well-known result E ( 2 n, 2 ) = 3 ζ ( 2 n ) / 4 was extended to E ( 2 n, 3 ) and E ( 2 n, 4 ) by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers E ( 2 n, k ) and then gave a direct formula for E ( 2 n, k ) for arbitrary k ≤ n . In this paper we apply a technique introduced by Granville to present an algorithm to calculate E ( 2 n, k ) and prove that the direct formula can also be deduced from Eisenstein's double product.