Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Congruences involving alternating multiple harmonic sums. | Congruences involving alternating multiple harmonic sums Roberto Tauraso Dipartimento di Matematica Universita di Roma Tor Vergata Italy tauraso@ Submitted Jun 4 2009 Accepted Jan 8 2010 Published Jan 14 2010 Mathematics Subject Classifications 11A07 11B65 05A19 Abstract We show that for any prime prime p 2 p-1 _ 1 k _ 1 cl2 g - -1 k 1 k 1 1 k mod p3 by expressing the left-hand side as a combination of alternating multiple harmonic sums. 1 Introduction In 8 Van Hamme presented several results and conjectures concerning a curious analogy between the values of certain hypergeometric series and the congruences of some of their partial sums modulo power of prime. In this paper we would like to discuss a new example of this analogy. Let us consider k 1 k1 1 i k I 1 1 1 3 1 1 3 5 1 1 3 5 7 k k 2 2 22i 3 rirf 4 FTT 1 c 1 dx 2 log x L 1 1 x -1 y 2log2. 2 Let p be a prime number what is the p-adic analogue of the above result The real case suggests to replace the logarithm with some p-adic function which behaves in a similar way. It turns out that the right choice is the Fermat quotient qp x xp-1 1 p THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R16 1 which is fine since qp x y qp x qp y mod p and as shown in 7 the following congruence holds for any prime p 2 P-1 1 7 7 k 1 k w 2qp 2 mod p . Here we improve this result to the following statement. Theorem . For any prime p 3 V - -11 k 2qp 2 - pqp 2 2 2p2qp 2 3 -7P2B1 3 12 p-1 2 1 k mod p3 k 1 where Bn is the n-th Bernoulli number. In the proof we will employ some new congruences for alternating multiple harmonic sums which are interesting in themselves such as H 1 2 p 1 y ij2 4Bp-3 mod p 0 i j p H 1 11 p 1 7 qp 2 3 gBp-3 mod p . ijk 8 0 i j k p J 2 Alternating multiple harmonic sums Let r 0 and let 1 a2 . ar G Z r. For any n r we define the alternating multiple harmonic sum as H a ao 0-n 17 sign ai ki H a1 a2 . . . ar n J I I ai 1 ki k2 --- kr n i 1 ki The integers r and Vr 1 ai are respectively the depth