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For k = 1,2,... let $H_k$ denote the harmonic number $∑_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have
$∑_{k=1}^{p-1} (H_k)/(k2^k) ≡ 7/24 pB_{p-3} (mod p²)$, $∑_{k=1}^{p-1} (H_{k,2})/(k2^k) ≡ - 3/8 B_{p-3} (mod p)$,
and
$∑_{k=1}^{p-1} (H²_{k,2n})/(k^{2n}) ≡ (\binom{6n+1}{2n-1} + n)/(6n+1) pB_{p-1-6n} (mod p²)$
for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.
$∑_{k=1}^{p-1} (H_k)/(k2^k) ≡ 7/24 pB_{p-3} (mod p²)$, $∑_{k=1}^{p-1} (H_{k,2})/(k2^k) ≡ - 3/8 B_{p-3} (mod p)$,
and
$∑_{k=1}^{p-1} (H²_{k,2n})/(k^{2n}) ≡ (\binom{6n+1}{2n-1} + n)/(6n+1) pB_{p-1-6n} (mod p²)$
for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}: = ∑_{j=1}^k 1/(j^m)$.
Słowa kluczowe
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Rocznik
Tom
Numer
Strony
67-78
Opis fizyczny
Daty
wydano
2013
Twórcy
autor
- Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
autor
- School of Mathematics, Hefei University of Technology, Hefei 230009, People's Republic of China
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-1-7