Symbolic summation methods and congruences involving harmonic numbers

Guo-Shuai Mao; Chen Wang; Jie Wang

doi:10.1016/j.crma.2019.10.005

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

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

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

Résumé
Abstract

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.

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.

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 :

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

@article{CRMATH_2019__357_10_756_0,
     author = {Guo-Shuai Mao and Chen Wang and Jie Wang},
     title = {Symbolic summation methods and congruences involving harmonic numbers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {756--765},
     publisher = {Elsevier},
     volume = {357},
     number = {10},
     year = {2019},
     doi = {10.1016/j.crma.2019.10.005},
     language = {en},
}

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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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