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Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering)  2007, Vol. 8 Issue (6): 946-948    DOI: 10.1631/jzus.2007.A0946
Computational Mathematics     
Congruences for finite triple harmonic sums
FU Xu-dan, ZHOU Xia, CAI Tian-xin
Department of Mathematics, Zhejiang University, Hangzhou 310028, China; 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)≡−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 pw+3; if w is odd, S(α,β,γ;p)≡rBpw (mod p) for pw, here r is an explicit rational number independent of p. A congruence of Catalan number is obtained as a special case.

Key wordsFinite triple harmonic sums      Recursive relation      Bernoulli numbers      Catalan numbers     
Received: 19 September 2006     
CLC:  O156  
Cite this article:

FU Xu-dan, ZHOU Xia, CAI Tian-xin. Congruences for finite triple harmonic sums. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 2007, 8(6): 946-948.

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http://www.zjujournals.com/xueshu/zjus-a/10.1631/jzus.2007.A0946     OR     http://www.zjujournals.com/xueshu/zjus-a/Y2007/V8/I6/946

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