Symbolic summation methods and congruences involving harmonic numbers

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

doi:10.1016/j.crma.2019.10.005

Number theory/Combinatorics

Symbolic summation methods and congruences involving harmonic numbers

Presented by: 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, Volume 357 (2019) no. 10, pp. 756-765.

Abstract
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.

Received: 2019-07-19
Accepted: 2019-10-04
Published online: 2019-10-01

DOI: 10.1016/j.crma.2019.10.005

Author's affiliations:

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

@article{CRMATH_2019__357_10_756_0,
     author = {Mao, Guo-Shuai and Wang, Chen and Wang, Jie},
     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},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2019.10.005/}
}

TY  - JOUR
AU  - Mao, Guo-Shuai
AU  - Wang, Chen
AU  - Wang, Jie
TI  - Symbolic summation methods and congruences involving harmonic numbers
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 756
EP  - 765
VL  - 357
IS  - 10
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2019.10.005/
DO  - 10.1016/j.crma.2019.10.005
LA  - en
ID  - CRMATH_2019__357_10_756_0
ER  -

%0 Journal Article
%A Mao, Guo-Shuai
%A Wang, Chen
%A Wang, Jie
%T Symbolic summation methods and congruences involving harmonic numbers
%J Comptes Rendus. Mathématique
%D 2019
%P 756-765
%V 357
%N 10
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2019.10.005/
%R 10.1016/j.crma.2019.10.005
%G en
%F CRMATH_2019__357_10_756_0

Mao, Guo-Shuai; Wang, Chen; Wang, Jie. 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://www.numdam.org/articles/10.1016/j.crma.2019.10.005/

References
Cited by

[1] Cao, H.-Q.; Pan, H. Note on some congruences of Lehmer, J. Number Theory, Volume 129 (2009) no. 8, pp. 1813-1819

[2] Hessami Pilehrood, Kh.; Hessami Pilehrood, T.; Tauraso, R. New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner's series, Trans. Amer. Math. Soc., Volume 366 (2014) no. 6, pp. 3131-3159

[3] Mao, G.-S.; Sun, Z.-W. Two congruences involving harmonic numbers with applications, Int. J. Number Theory, Volume 12 (2016) no. 2, pp. 527-539

[4] Mao, G.-S. Proof of two conjectural supercongruences involving Catalan–Larcombe–French numbers, J. Number Theory, Volume 179 (2017), pp. 88-96

[5] Mao, G.-S. Proof of some congruences conjectured by Z.-W. Sun, Int. J. Number Theory, Volume 13 (2017), pp. 1983-1993

[6] Osburn, R.; Schneider, C. Gaussian hypergeometric series and supercongruences, Math. Comput., Volume 78 (2009), pp. 275-292

[7] Schneider, C. Symbolic summation assists combinatorics, Sémin. Lothar. Comb., Volume 56 (2007)

[8] Sun, Z.-H. Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., Volume 105 (2000), pp. 193-223

[9] Sun, Z.-H. Congruences involving Bernoulli and Euler numbers, J. Number Theory, Volume 128 (2008) no. 2, pp. 280-312

[10] Sun, Z.-W. Super congruences and Euler numbers, Sci. China Math., Volume 54 (2011), pp. 2509-2535

[11] Sun, Z.-W. p-adic congruences motivated by series, J. Number Theory, Volume 134 (2014), pp. 181-196

[12] Sun, Z.-W. A new series for $π^{3}$ and related congruences, Internat. J. Math., Volume 26 (2015) no. 8

[13] Sun, Z.-W.; Zhao, L.-L. Arithmetic theory of harmonic numbers (II), Colloq. Math., Volume 130 (2013) no. 1, pp. 67-78

[14] Tauraso, R. Supercongruences related to ${}_{3}F_{2} (1)$ involving harmonic numbers, Int. J. Number Theory, Volume 14 (2018) no. 4, pp. 1093-1109

[15] Wolstenholme, J. On certain properties of prime numbers, Q. J. Math., Volume 5 (1862), pp. 35-39

[16] Zhao, L.-L.; Sun, Z.-W. Some curious congruences modulo primes, J. Number Theory, Volume 130 (2010) no. 4, pp. 930-935

Cited by Sources: