arXiv subjects
Publications and source records attributed to Farid Bencherif.
In this paper, we establish an identity for Bernoulli's generalized polynomials. We deduce generalizations for many relations involving classical Bernoulli numbers or polynomials. In particular, we generalize a recent Gessel identity.
The Bernoulli numbers are fascinating and ubiquitous numbers, they occur in several domains of Mathematics like Number theory (FLT), Group theory, Calculus and even in Physics. Since Bernoulli's work, they are yet studied to understand their deep nature and particularly to find relationships between them. In this paper, we give, firstly, a short response to a problem stated, in 1971, by Carlitz and studied by many authors like Prodinger \cite{PRO}, the second pearl is an answer to a question raised, in 2008, by Tom Apostol .The third pearl is another proof of a relationship already given in 2011, by the authors.
In this paper, we prove that for any odd prime $p$ and for any $p$-integer $α$,we have $ \binom{αp-1}{p-1}\equiv 1-α(α-1)(α^{2}-α-1)p\sum_{k=1}^{p-1}\frac{1}{k}+α^{2} (α-1)^{2}p^{2}\sum_{1\leq i<j\leq p-1}\frac{1}{ij} \ \pmod{p^{m}}$, where $m=7$ if $p\neq 7$ and $m=6$ if $p=7$. this congruence generalizes the congruences of Wolstenholme, Morley, Glaisher, Carlitz, McIntosh, Tauraso and Meštrović. It allows one to rediscover the congruences of Glaisher, Carlitz and Zhao in a simple way
In this paper, we give a short proof of a relation generalizing many identities for Bernoulli numbers.
In this article, we give a positive answer to a question posed in 1960 by D.S. Mitrinović and R.S. Mitrinović (see: D.S. Mitrinović et R.S. Mitrinović, Tableaux qui fournissent des polynômes de Stirling, Publications de la Faculté d'Electronique, série: Mathématiques et physique, 34, (1960).1-23.) concerned the Stirling numbers of the first kind $s(n,k).$ We prove that for all $k\geq 2$ there exist an integer $m_{k}$ and a primitive polynomial $P_{k}(x)$ in $\mathbb{Z}[x]$ such that for all $n\geq k$, $s(n,n-k)=\frac{1}{m_{k}}\binom{n}{k+1}\left(n(n-1)\right) ^{\mathop{\rm mod}\nolimits (k,2)}P_{k}(n)$. Moreover for all $k\geq1$, $P_{2k}(0)=P_{2k+1}(0)$.
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations.
In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Cerin and Z. Cerin & G. M. Gianella, more easily.