Symbolic summation methods and congruences involving harmonic numbers

Mao, Guo-Shuai; Wang, Chen; Wang, Jie

doi:10.1016/j.crma.2019.10.005

Geodesic

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Number theory/Combinatorics

Symbolic summation methods and congruences involving harmonic numbers
[Méthodes de sommation symbolique et congruences impliquant les nombres harmoniques]

Présenté par : the Editorial Board

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

Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 756-765.

Voir la notice de l'article provenant de la source Numdam

Abstract (VO)
Résumé

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

\sum_{k = 0}^{(p - 3) / 2} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{(2 k + 1) 16^{k}} H_{k}^{(2)} \equiv - 7 B_{p - 3} (\mod p),

\sum_{k = 1}^{p - 1} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{k 16^{k}} H_{2 k}^{(2)} \equiv B_{p - 3} (\mod p),

\sum_{k = 1}^{(p - 1) / 2} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{k 16^{k}} (H_{2 k} - H_{k}) \equiv - \frac{7}{3} p B_{p - 3} (\mod p^{2}),

where

H_{n}^{(m)} = \sum_{k = 1}^{n} 1 / k^{m}

(

m \in Z^{+} = {1, 2, \dots}

) is the n-th harmonic numbers of order m and

B_{n}

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

\sum_{k = 0}^{(p - 3) / 2} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{(2 k + 1) 16^{k}} H_{k}^{(2)} \equiv - 7 B_{p - 3} (\mod p),

\sum_{k = 1}^{p - 1} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{k 16^{k}} H_{2 k}^{(2)} \equiv B_{p - 3} (\mod p),

\sum_{k = 1}^{(p - 1) / 2} \frac{{(\begin{matrix} 2 k \\ k \end{matrix})}^{2}}{k 16^{k}} (H_{2 k} - H_{k}) \equiv - \frac{7}{3} p B_{p - 3} (\mod p^{2}),

où

H_{n}^{(m)} = \sum_{k = 1}^{n} 1 / k^{m}

(

m \in {1, 2, \dots}

) désigne le n-ième nombre harmonique d'ordre m et

B_{n}

est le n-ième nombre de Bernoulli.

Reçu le : 2019-07-19
Accepté le : 2019-10-04
Publié le : 2019-10-01

DOI : 10.1016/j.crma.2019.10.005

Affiliations des auteurs :

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

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     title = {Symbolic summation methods and congruences involving harmonic numbers},
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Mao, Guo-Shuai; Wang, Chen; Wang, Jie. Symbolic summation methods and congruences involving harmonic numbers. Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 756-765. doi : 10.1016/j.crma.2019.10.005. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2019.10.005/

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