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Bakir Farhi

Publications and source records attributed to Bakir Farhi.

At least 19 recordsLinked to original sources
arXiv2026

A measure of intelligence of an approximation of a real number in a given model

In this paper, we introduce a way to measure the intelligence (or relevance) of an approximation of a given real number in a given model of approximation. Based on the notion of complexity of a number, defined as the number of its digits (in a given base), we introduce a function noted $μ$ (called a measure of intelligence) that associates to any approximation $\mathbf{app}$ of a given real number in a given model a positive number $μ(\mathbf{app})$, which measures the quality of that approximation. More precisely, an approximation $\mathbf{app}$ is deemed intelligent if and only if $μ(\mathbf{app}) \geq 1$. We illustrate the theory with several numerical examples and apply it to the rational model. In this case, we show that it is consistent with the classical theory of rational Diophantine approximation. We conclude by stating an open problem, namely whether any real number can be intelligently approximated in a given model for which it is a limit point.

math.GM↗
arXiv2026

On refinements of two-term Machin-like formulas

We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \fracπ{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $α:= \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \fracπ{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $α$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $π/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = π/4$.

math.NT↗
arXiv2025

A polynomial approach to Carlitz's $q$-Bernoulli numbers

This paper investigates $q$-analogues of the classical Bernoulli polynomials and numbers. We introduce a new polynomial sequence ${\left(B_{n , q}(X)\right)}_{n \in \mathbb{N}_0}$, defined via the Jackson integral, and explore its connections with Carlitz's $q$-Bernoulli polynomials and numbers. Specifically, we prove that the numbers $B_{n , q}(0)$ are exactly the Carlitz $q$-Bernoulli numbers and that the polynomials $B_{n , q}(X)$ are genuine $q$-analogues of the classical Bernoulli polynomials. This approach leverages the Jackson integral to reformulate Carlitz's $q$-Bernoulli numbers in terms of classical polynomial structures, offering new insights into their properties.

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arXiv2025

Expansion of a bivariate symmetric mean in the neighborhood of the first bisector

In this paper, we investigate the behavior of a bivariate mean $M$ near the first bisector by establishing, in several significant cases, an important expansion of $M$ derived from the Taylor expansion of a single-variable function. These expansions are made explicit for a number of classical means. This motivates the introduction of the concept of the characteristic function $Q_M$ of a mean $M$, defined as the second partial derivative of $M$ with respect to its first variable, evaluated along the diagonal. The function $Q_M$ measures the proximity of $M$ to the arithmetic mean near the first bisector and provides a univariate analytic framework for comparing and classifying means. We prove that inequalities between characteristic functions yield local inequalities between the corresponding means, and that in the case of homogeneous means, such inequalities hold globally. We also examine several important classes of means, both classical and novel, including: normal means, additive means, integral means of the first kind, integral means of the second kind, weighted integral means of the first kind, and weighted integral means of the second kind. For each class, we determine the specific form taken by the characteristic functions $Q_M$ of the means $M$ it contains, and we then study the injectivity and the surjectivity of the mapping $M \mapsto Q_M$ within the class. We also use characteristic functions to investigate intersections between certain classes of means, highlighting one of the key strengths of this concept. Finally, we introduce and study, for a given mean $M$, the class of $M$-means, and show, in particular, that the arithmetic-geometric mean $\mathrm{AGM}$ is an $M$-mean for a specific weighted integral mean of the first kind $M$.

math.NT↗
arXiv2025

$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials

In this paper, we investigate a specific class of $q$-polynomial sequences that serve as a $q$-analogue of the classical Appell sequences. This framework offers an elegant approach to revisiting classical results by Carlitz and, more interestingly, to establishing an important extension of the Carlitz $q$-Bernoulli polynomials and numbers. In addition, we establish explicit series representations for our extended Carlitz $q$-Bernoulli numbers and express them in terms of $q$-Stirling numbers of the second kind. This leads to a novel formula that explicitly connects the Carlitz $q$-Bernoulli numbers with the $q$-Stirling numbers of the second kind.

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arXiv2023

Characterization of the Bernoulli polynomials via the Raabe functional equation

The purpose of the present paper is to show that in certain classes of real (or complex) functions, the Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of the real $1$-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not relating to the Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich.

math.CA↗
arXiv2022

Nontrivial lower bounds for the $p$-adic valuations of some type of rational numbers and an application for establishing the integrality of some rational sequences

In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.

math.NT↗
arXiv2022

Nontrivial lower bounds for the $p$-adic valuations of some type of rational numbers

In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both related to the special case $p = 2$. The crucial point for obtaining our main result is the fact that the $p$-adic valuation of the rational numbers in question is unbounded from above. We will confirm this fact by three different methods; the first two are elementary while the third one leans on the $p$-adic analysis.

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arXiv2022

On some products taken over the prime numbers

This paper is devoted to study some expressions of the type $\prod_{p} p^{\lfloor\frac{x}{f(p)}\rfloor}$, where $x$ is a nonnegative real number, $f$ is an arithmetic function satisfying some conditions, and the product is over the primes $p$. We begin by proving that such expressions can be expressed by using the $\mathrm{lcm}$ function, without any reference to prime numbers; we illustrate this result with several examples. The rest of the paper is devoted to study the two particular cases related to $f(m) = m$ and $f(m) = m - 1$. In both cases, we found arithmetic properties and analytic estimates for the underlying expressions. We also put forward an important conjecture for the case $f(m) = m - 1$, which depends on the counting of the prime numbers of a special form.

math.NT↗
arXiv2022

The integrality of the Genocchi numbers obtained through a new identity and other results

In this note, we investigate some properties of the integer sequence of general term $a_n := \sum_{k = 0}^{n - 1} k! (n - k - 1)!$ ($\forall n \geq 1$) to derive a new identity of the Genocchi numbers $G_n$ ($n \in \mathbb{N}$), which immediately shows that $G_n \in \mathbb{Z}$ for any $n \in \mathbb{N}$. In another direction, we obtain nontrivial lower bounds for the $2$-adic valuations of the rational numbers $\sum_{k = 1}^{n} \frac{2^k}{k}$.

math.NT↗
arXiv2022

On a curious integer sequence

This note is devoted to study the recurrent numerical sequence defined by: $a_0 = 0$, $a_n = \frac{n}{2} a_{n - 1} + (n - 1)!$ ($\forall n \geq 1$). Although, it is immediate that ${(a_n)}_n$ is constituted of rational numbers with denominators powers of $2$, it is not trivial that ${(a_n)}_n$ is actually an integer sequence. In this note, we prove this fact by expressing $a_n$ in terms of the Genocchi numbers and the Stirling numbers of the first kind. We derive from our main result several corollaries and we conclude with some remarks and open problems.

math.NT↗
arXiv2021

On the sum of the values of a polynomial at natural numbers which form a decreasing arithmetic progression

The purpose of this paper consists to study the sums of the type $P(n) + P(n - d) + P(n - 2 d) + \dots$, where $P$ is a real polynomial, $d$ is a positive integer and the sum stops at the value of $P$ at the smallest natural number of the form $(n - k d)$ ($k \in \mathbb{N}$). Precisely, for a given $d$, we characterize the $\mathbb{R}$-vector space ${\mathscr{E}}_d$ constituting of the real polynomials $P$ for which the above sum is polynomial in $n$. The case $d = 2$ is studied in more details. In the last part of the paper, we approach the problem through formal power series; this inspires us to generalize the spaces $\mathscr{E}_d$ and the underlying results. Also, it should be pointed out that the paper is motivated by the curious formula: $n^2 + (n - 2)^2 + (n - 4)^2 + \dots = \frac{n (n + 1) (n + 2)}{6}$, due to Ibn al-Banna al-Marrakushi (around 1290).

math.NT↗
arXiv2021

Arithmetic properties of the Genocchi numbers and their generalization

This note is devoted to establish some new arithmetic properties of the generalized Genocchi numbers $G_{n , a}$ ($n \in \mathbb{N}$, $a \geq 2$). The resulting properties for the usual Genocchi numbers $G_n = G_{n , 2}$ are then derived. We show for example that for any even positive integer $n$, the Genocchi number $G_n$ is a multiple of the odd part of $n$.

math.NT↗
arXiv2020

A new generalization of the Genocchi numbers and its consequence on the Bernoulli polynomials

This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by ${(B_n)}_{n \in \mathbb{N}}$ the sequence of the Bernoulli numbers and by ${(B_n(X))}_{n \in \mathbb{N}}$ the sequence of the Bernoulli polynomials, we especially obtain that for any natural number $n$, the reciprocal polynomial of the polynomial $\big(B_n(X) - B_n\big)$ is integer-valued.

math.NT↗
arXiv2020

Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression

This paper is devoted to establish nontrivial effective lower bounds for the least common multiple of consecutive terms of a sequence ${(u_n)}_{n \in \mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q , r$ are positive integers and $u_0$ is a non-negative integer such that $\mathrm{gcd}(u_0 , r) = \mathrm{gcd}(u_1 , q) = 1$. For such a sequence, we show that for all positive integer $n$, we have $\mathrm{lcm}\{u_1 , u_2 , \dots , u_n\} \geq c_1 \cdot c_2^n \cdot q^{\frac{n^2}{4}}$, where $c_1$ and $c_2$ are positive constants depending only on $q , r$ and $u_0$. This can be considered as a $q$-analog of the lower bounds already obtained by the author (in 2005) and by Hong and Feng (in 2006) for the arithmetic progressions.

math.NT↗
arXiv2020

Identities and estimations involving the least common multiple of strong divisibility sequences

In this paper, we first prove that for any strong divisibility sequences $\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity: $\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots, \binom{n}{n}_{\bf{a}} \right\rbrace = \frac{\mathrm{lcm} \left(a_1,\dots , a_n , a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \geq 1\right)$, generalizing the identity of Farhi (obtained in 2009 for $a_n=n$). Then, we derive from this one some other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some Lucas sequences. Denoting by $\left(F_n\right)_n$ the usual Fibonacci sequence, we prove for example that for all $n \geq 1$, we have \[ Φ^{\frac{n^2}{4}-\frac{9}{4}} \leq \mathrm{lcm}\left(F_1,\dots,F_n\right) \leq Φ^{\frac{n^2}{3}+\frac{4n}{3}} , \] where $Φ$ denotes the golden ratio.

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