Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: EVALUATION OF TRIPLE EULER SUMS. | EVALUATION OF TRIPLE EULER SUMS Jonathan M. Borwein1 CECM Department of Mathematics and Statistics Simon Fraser University Burnaby . V5A 1S6 Canada e-mail jborwein@ Roland Girgensohn Institut fur Mathematik Medizinische Universităt Lubeck D-23560 Lubeck Germany e-mail girgenso@ Submitted December 16 1995 Accepted August 7 1996 Abstract. Let a b c be positive integers and define the so-called triple double and single Euler sums by 1 x 1y 1 1 xaybzc i a M J2J2J2 x 1 y 1 z 1 1 x 1 1 c a b 2 h and a E xa x y x Extending earlier work about double sums we prove that whenever a b c is even or less than 10 then a b c can be expressed as a rational linear combination of products of double and single Euler sums. The proof involves finding and solving linear equations which relate the different types of sums to each other. We also sketch some applications of these results in Theoretical Physics. Introduction This paper is concerned with the discussion of sums of the type c a b c x 1y 1z 1 xaybzc For which values of the integer parameters a b c can these sums be expressed in terms of the simpler series 1 x 1 1 oc c a b Ỹ Xỹ and c a E - x 1y 1 y x 1 1 Research supported by NSERC and the Shrum Endowment of Simon Fraser University. AMS 1991 subject classification Primary 40A25 40B05 Secondary 11M99 33E99. Key words Riemann zeta function Euler sums polylogarithms harmonic numbers quantum field theory knot theory THE ELECTRONIC JOURNAL OF COMBINATORICS 3 1996 R23 2 We call sums of this type triple double or single Euler sums because Euler was the first to find relations between them cf. 9 of course the single Euler sums are values of the Riemann zeta function at integer arguments . Investigation of Euler sums has a long history. Euler s original contribution was a method to reduce double sums to certain rational linear combinations of products of single sums. Examples for such evaluations all due to Euler are c 2 1 c 3 c 3 1 3 1 4 -- 2c4 - 1 c2
