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Congruences for finite triple harmonic sums
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FU Xu-dan, ZHOU Xia, CAI Tian-xin. Congruences for finite triple harmonic sums[J]. Journal of Zhejiang University Science A, 2007, 8(6): 946-948.
@article{title="Congruences for finite triple harmonic sums",
author="FU Xu-dan, ZHOU Xia, CAI Tian-xin",
journal="Journal of Zhejiang University Science A",
volume="8",
number="6",
pages="946-948",
year="2007",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2007.A0946"
}
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T1 - Congruences for finite triple harmonic sums
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DOI - 10.1631/jzus.2007.A0946
Abstract: Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡−2Bp−3 (mod p), which holds when p=3 evidently. In this paper, we consider finite triple harmonic sum S(α,β,γ;p) (mod p) is considered for all positive integers α,β,γ. We refer to w=α+β+γ as the weight of the sum, and show that if w is even, S(α,β,γ;p)≡0 (mod p) for p≥w+3; if w is odd, S(α,β,γ;p)≡rBp−w (mod p) for p≥w, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.
[1] Graham, R.L., Knuth, D.E., Patashnik, O., 1994. Concrete Mathematics (2nd Ed.). Addison-Wesley.
[2] Hoffman, M.E., 2004. Quasi-symmetric Functions and Mod p Multiple Harmonic Sums. Http://arxiv.org/abs/math.NT/0401319
[3] Ji, C.G., 2005. A simple proof of a curious congruence by Zhao. Proc. Amer. Math. Soc., 133:3469-3472.
[4] Zhao, J.Q., 2003a. Bernoulli Numbers, Wolstenholme’s Theorem, and p5 Variations of Lucas’ Theorem. Http://arxiv.org/abs/math.NT/0303332, V1.
[5] Zhao, J.Q., 2003b. Partial Sums of Multiple Zeta Value Series I: Generalizations of Wolstenholme’s Theorem. Http://arxiv.org/abs/math.NT/0301252, V2.
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