Báo cáo toán học: "Congruences involving alternating multiple harmonic sums"

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

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