| Congruences for finite triple harmonic sums |
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| 作者姓名： | FU Xu-dan ZHOU Xia CAI Tian-xin |
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| 作者单位： | FU Xu-dan1，2，ZHOU Xia1，CAI Tian-xin1 (1Department of Mathematics，Zhejiang University，Hangzhou 310028，China) (2Hangzhou Foreign Language School，Hangzhou 310023，China) |
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| 基金项目： | Project (No. 10371107) supported by the National Natural Science Foundation of China |
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| 摘 要： | 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.
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| 关 键 词： | 有限三重调和和 同余方程 回归关系 柏努利数 |
| 收稿时间： | 2006-09-19 |
| 修稿时间： | 2007-01-18 |
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,2007,8(6):946-948. |
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| Authors: | Fu Xu-dan Zhou Xia Cai Tian-xin |
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| Institution: | (1) Department of Mathematics, Zhejiang University, Hangzhou, 310028, China;(2) Hangzhou Foreign Language School, Hangzhou, 310023, China |
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| Abstract: | Zhao (2003a) first established a congruence for any odd prime p>3, S(1,1,1;p)≡−2B
p−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,γ,γ,p)≡rB
p−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.
Project (No. 10371107) supported by the National Natural Science Foundation of China |
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| Keywords: | Finite triple harmonic sums Recursive relation Bernoulli numbers Catalan numbers |
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