Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k. | Evaluations of fc-fold Euler Zagier sums a compendium of results for arbitrary k J. M. Borwein jborwein@ D. M. Bradley dbradley@ CECM Simon Fraser University Burnaby . V5A 1S6 Canada http D. J. Broadhurst Physics Department Open University Milton Keynes MK7 6AA UK http Submitted September 2 1996 Accepted October 31 1996. Abstract. Euler sums also called Zagier sums occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here we assemble results for Euler Zagier sums also known as multidimensional zeta harmonic sums of arbitrary depth including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking we have sketched derivations of all results that have so far been proved. THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 1997 R5 2 1 Introduction We consider k-fold Euler sums 13 2 3 also called Zagier sums of arbitrary depth k. These sums occur in a natural way within the context of knot theory and quantum field theory see 4 for an extended bibliography carrying on a rich tradition of algebra and number theory as pioneered by Euler. There are various conjectures related to these sums see . 8 below whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field whence new results are welcome. As in 4 we allow for all possible alternations of signs with ơj 1 in X Y ơ c s1 . sk ơ1 . ơk II 1 nj nj 1 0 j 1 nj since alternating Euler sums are essential 7 to the connection 18 of knot theory with quantum field theory 8 6 .
