Laid Elkhiri, Miloud Mihoubi and Abdellah Derbal

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 4, Pages 39–51

DOI: 10.7546/nntdm.2020.26.4.39-51

**Full paper (PDF, 187 Kb)**

## Details

### Authors and affiliations

Laid Elkhiri

*Ibn Khaldoun University, EDPNLHM Laboratory of ENS
Old Kouba, Tiaret, Algeria
*

Miloud Mihoubi

*USTHB, Faculty of Mathematics, RECITS Laboratory
PO Box 32 16111 El-Alia, Bab-Ezzouar, Algiers, Algeria
*

Abdellah Derbal

*ENS, Department of Mathematics, EDPNLHM Laboratory
Old Kouba, Algiers, Algeria
*

### Abstract

In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number (super congruences) in the ring of -integer involving binomial coefficients and generalized harmonic numbers.

### Keywords

- Binomial coefficients
- Harmonic numbers
- Generalized harmonic numbers
- Congruences

### 2010 Mathematics Subject Classification

- 11A07
- 11B65
- 11B99

### References

- Choi, J., & Srivastava, H. M. (2011). Some summation formulas involving harmonic numbers and generalized harmonic numbers, Mathematical and computer Modelling, 54, 2220–2234.
- Dilcher, K. (1995). Some
*q*-series identities related to divisor factors, Discrete Math., 145, 83–93. - Glaisher, J.W.L. (1900). A general congruence theorem relating to the Bernoullian function, Proc. London Math. Soc., 33, 27–56.
- Gould, H. W. (1972). Combinatorial Identities. A Standardized Set of Tables Listing 500 Binomial Coefficient Summations, Morgantown, W. Va.
- He, B. (2017). Some congruences on harmonic numbers and binomial sums, Period. Math. Hung., 74, 67–72.
- Koparal, S., & Omur, N. (2016). Congruences related to central binomial coefficients, harmonic and Lucas numbers, Turk. J. Math., 40, 973–985.
- Lehmer, E. (1938). On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math., 39, 350–360.
- Mestrovic, R., & Andji, M. (2017). Certain congruence for harmonic numbers, Mathematica Montisnigri, 38.
- Morley, F. (1895). Note on the congruence 2
^{4n}≡ (−1)^{n}(2*n*)!= (*n*!)^{2}, where 2*n*+ 1 is a prime, Ann. Math., 9(1–6), 168–170. - Prodinger, H. (2000). A
*q*-analogue of a formula of Hernandez obtained by inverting a result of Dilcher, Australas. J. Comb., 21, 271–274. - Sun, Z. H. (2000). Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105, 193–223.
- Sun, Z. H. (2008). Congruences involving Bernoulli and Euler numbers, J. Number Theory, 128, 280–312.
- Sun, Z. W. (2015). Super congruences motivated by
*e*, J. Number Theory, 147, 326–341. - Sun, Z. W., & Zhao, L. L. (2013). Arithmetic theory of Harmonic numbers (II), Colloq. Math., 130(1), 67–78.
- Wang, W. S. (2015). Sums of involving the Harmonic numbers and the Binomial

coefficients, American Journal of Computational Mathematics, 5, 96–105.

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## Cite this paper

Elkhiri, L., Mihoubi, M., & Derbal, A. (2020). Congruences involving alternating sums related to harmonic numbers and binomial coefficients, 26(4), 39-51, DOI: 10.7546/nntdm.2020.26.4.39-51.