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12 results on '"ELKHIRI, LAID"'

3. Super-congruences involving trininomial coefficients

Author

Elkhiri, Laid and Mihoubi, Miloud

Subjects
Mathematics - Number Theory, 11B65, 11A07, 05A10
Abstract

The aim of this work is to establish congruences $\left( \operatorname{mod}p^{2}\right) $ involving the trinomial coefficients $\binom{np-1}{p-1}_{2}$ and $\binom{np-1}{\left( p-1\right)/2}_{2}$ arising from the expansion of the powers of the polynomial $1+x+x^{2}.$ In main results we extend some known congruences involving the binomial coefficients $\binom{np-1}{p-1}$ and $\binom{np-1}{\left( p-1\right) /2}$ and establish congruences link binomial coefficients and harmonic numbers.

Published
2019

5. Some congruences with q--binomial sums.

Author

ÖMÜR, Neşe, GÜR, Zehra Betül, KOPARAL, Sibel, and ELKHİRİ, Laid

Subjects
INTEGERS, GEOMETRIC congruences
Abstract

In this paper, using some combinatorial identities and congruences involving q--harmonic numbers, we establish congruences that for any odd prime p and any positive integer α, ... and ... when n is any integer. [ABSTRACT FROM AUTHOR]

Published
2023
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7. Some congruences with q−binomial coefficients and q− harmonic numbers

Author

KOPARAL, Sibel, ÖMÜR, Neşe, and ELKHİRİ, Laid

Subjects
Matematik, congruences, q−binomial coefficients, q−harmonic numbers, Abel sum, Mathematics
Abstract

\begin{abstract}In this paper, considering $q-$analogues and $q-$combinatorial identities,we gave some congruences including $q-$binomial coefficients and $q-$%harmonic numbers. For example, for any prime number $p$ and $\alpha \in%TCIMACRO{\U{2124} }%%BeginExpansion\mathbb{Z}%EndExpansion^{+},$%\begin{eqnarray*}&&\sum\limits_{k=1}^{p-1}\left( -1\right) ^{k}q^{-\alpha pk+\binom{k+1}{2}+k}\left[ k\right] _{q}{\alpha p-1 \brack k}_{q} \\&\equiv &\frac{q^{1-\alpha p}}{(1-q^{2})^{2}}\left( q^{\alpha p+2}\left(q^{p}-2\right) +q^{\alpha p}-q^{p}+q^{2}\right) \left[ p-1\right] _{q} %\pmod{\left[ p\right] _{q}^{3}}.\end{eqnarray*}\end{abstract}

Published
2022

11. A Q-ANALOGUE OF GRANVILLE'S CONGRUENCE.

Author

Elkhiri, Laid, Ömür, Neşe, and Koparal, Sibel

Subjects
ODD numbers, REAL numbers, GEOMETRIC congruences
Abstract

In this paper, we establish a q-analogue of Granville's congruence as follows: ... for any real number x and odd prime p: Here [n]q = 1 + q + q² + ... + qn-1 and (x; q)n = (1-x) (1 - xq)... (1- xqn-1). [ABSTRACT FROM AUTHOR]

Published
2022

12. CONGRUENCES INVOLVING SUMS OF HARMONIC NUMBERS AND BINOMIAL COEFFICIENTS.

Author

ELKHIRI, LAID, MIHOUBI, MILOUD, and DERBAL, ABDELLAH

Subjects
GEOMETRIC congruences, BINOMIAL coefficients, MATHEMATICIANS, CATALAN numbers, MATHEMATICAL analysis
Abstract

Congruences involving sums of Harmonic numbers and binomial coefficients are considered in this paper. Recently, many great mathematicians have been interested to find congruences and relationships between these numbers such Sun & Tauraso, Koparal & Ömür, Mao & Sun and Meštrović & Andjić. In the present paper, some new combinatorial congruences are proved. These congruences are mainly determined modulo p² or p³ (in any prime) and they are motivated by a recent paper by Meštrović and Andjić. The first main result (Theorem 1) presents the congruence modulo p² (p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. Two interesting congruences modulo a prime p > 3 (Corollary 2) involving Harmonic numbers Hk, Catalan numbers ck and Fermat quotient q2 := (2p-1 - 1)/p are obtained as consequences of Theorem 1. The second main result (Theorem 2) presents the congruence modulo p³(p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. [ABSTRACT FROM AUTHOR]

Published
2020
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