Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Combinatorial Aspects of Multiple Zeta Values. | Combinatorial Aspects of Multiple Zeta Values Jonathan M. Borwein1 CECM Department of Mathematics and Statistics Simon Fraser University Burnaby . V5A 1S6 Canada e-mail jborwein@ David M. Bradley2 Department of Mathematics and Statistics Dalhousie University Halifax . B3H 3J5 Canada e-mail bradley@ David J. Broadhurst Physics Department Open University Milton Keynes MK7 6AA UK e-mail Petr Lisonek3 CECM Department of Mathematics and Statistics Simon Fraser University Burnaby . V5A 1S6 Canada e-mail lisonek@ Submitted July 2 1998 Accepted August 1 1998. 1 Research supported by NSERC and the Shrum Endowment of Simon Fraser University. 2Work done while the author was recipient of the NSERC Postdoctoral Fellowship. 3Industrial Postdoctoral Fellow of PIms The Pacific Institute for the Mathematical Sciences . AMS 1991 subject classification Primary 05A19 11M99 68R15 Secondary 11Y99. Key words Multiple zeta values Euler sums Zagier sums factorial identities shuffle algebra. 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 5 1998 R38 2 Abstract Multiple zeta values MZVs also called Euler sums or multiple harmonic series are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity. 1 Introduction In this paper we continue our study of multiple zeta values MZVs sometimes also called Euler sums or Zagier sums defined by k c si . Sk IK ni n2 --- nfc 0 j 1 with Sj 2 z and s1 1 to ensure the convergence. The integer k is called the depth
