Number theory/Combinatorics
Symbolic summation methods and congruences involving harmonic numbers
Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 756-765.

In this paper, we establish some combinatorial identities involving harmonic numbers via the package Sigma, by which we confirm some conjectural congruences of Z.-W. Sun. For example, for any prime $p>3$, we have

 $∑k=0(p−3)/2(2kk)2(2k+1)16kHk(2)≡−7Bp−3(modp),$
 $∑k=1p−1(2kk)2k16kH2k(2)≡Bp−3(modp),$
 $∑k=1(p−1)/2(2kk)2k16k(H2k−Hk)≡−73pBp−3(modp2),$
where $Hn(m)=∑k=1n1/km$ ($m∈Z+={1,2,…}$) is the n-th harmonic numbers of order m and $Bn$ is the n-th Bernoulli number.

Nous montrons ici, à l'aide du progiciel Sigma, quelques identités combinatoires faisant intervenir les nombres harmoniques. Nous établissons ainsi des congruences conjecturées par Z.-W. Sun. Par exemple, pour $p>3$ premier, on a

 $∑k=0(p−3)/2(2kk)2(2k+1)16kHk(2)≡−7Bp−3(modp),$
 $∑k=1p−1(2kk)2k16kH2k(2)≡Bp−3(modp),$
 $∑k=1(p−1)/2(2kk)2k16k(H2k−Hk)≡−73pBp−3(modp2),$
$Hn(m)=∑k=1n1/km$ ($m∈{1,2,…}$) désigne le n-ième nombre harmonique d'ordre m et $Bn$ est le n-ième nombre de Bernoulli.

Accepted:
Published online:
DOI: 10.1016/j.crma.2019.10.005

Guo-Shuai Mao 1; Chen Wang 2; Jie Wang 2

1 Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People's Republic of China
2 Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
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Guo-Shuai Mao; Chen Wang; Jie Wang. Symbolic summation methods and congruences involving harmonic numbers. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 756-765. doi : 10.1016/j.crma.2019.10.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.005/

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