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Fethi Bouzeffour

Publications and source records attributed to Fethi Bouzeffour.

17 recordsLinked to original sources
arXiv2024

Mehta's eigenvectors for the finite Hartely transform

This paper presents a novel approach for evaluating analytical eigenfunctions of the finite Hartley transform. The approach is based on the use of $N=1/2$-supersymmetric quantum mechanics as a fundamental tool, which builds on the key observation that the Hartley transform commutes with the supercharge operator. Using the intertwining operator between the Hartley transform and the finite Hartley transform, our approach provides an overcomplete basis of eigenvectors expressed in terms of supersymmetric Hermite polynomials.

quant-ph↗
arXiv2024

Meixner $d$-Orthogonal Polynomials Arising from $\mathfrak{su}(1,1)$

In this study, we present a novel family of Meixner-type $d$-orthogonal polynomials, which are distinguished as a particular subset of multiple orthogonal polynomials. We demonstrate their connection to the Lie algebra $\mathfrak{su}(1,1)$ by identifying them as matrix elements of an appropriately defined nonlinear operator. Utilizing Barut-Girardello coherent states, we explicitly outline their key features, including recurrence relations, generating functions, and $d$-orthogonality relations, among others.

math.CA↗
arXiv2019

$C_λ$- Extended oscillator algebra and $d$-orthogonal polynomials

In this paper we first construct an analytic realization of the $C_λ$-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of $d$-orthogonal polynomials which are extensions of the Hermite and Laguerre polynomials. The underlying algebraic framework allowed us a systematic derivation of their main properties such as recurrence relations, difference-differential equations, lowering and rising operators and generating functions. Finally, we use these polynomials to construct a realization of the $C_λ$-extended oscillator by block matrices.

math-ph↗
arXiv2018

Arithmetical properties at the level of idempotence

In this paper we give an attempt to extend some arithmetic properties such as multiplicativity, convolution products to the setting of operators theory. We provide a significant examples which are of interest in number theory. We also give a representation of the Euler differential operator by means of the Euler totient arithmetic function and idempotent elements of some associative unital algebra.

math.CA↗
arXiv2018

Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey-Wilson and of the little q-Jacobi case.

math.CA↗
arXiv2016

Functional limit theorems for the number of busy servers in a $G/G/\infty$ queue

We discuss weak convergence of the number of busy servers in a $G/G/\infty$ queue in the $J_1$-topology on the Skorokhod space. We prove two functional limit theorems, with random and nonrandom centering, respectively, thereby solving two open problems stated in Mikosch and Resnick (2006}. A new integral representation for the limit Gaussian process is given.

math.PR↗
arXiv2016

Jackson's $(-1)$-Bessel functions with the Askey-Wilson algebra setting

The aim of this work is to study new functions arising from the limit transition of the Jackson's $q$-Bessel functions when $q\rightarrow -1$. These functions coincide with the $cas$ function for particular values of their parameters. We prove also that these functions are eigenfunction of differential-difference operators of Dunkl-type. Further, we consider special cases of the Askey-Wilson algebra $AW(3)$ that have these operators (up to constants) as one of their three generators and whose defining relations are given in terms of anticommutators.

math.CA↗
arXiv2016

A law of the iterated logarithm for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is an infinite occupancy scheme obtained by allocating the points of a uniform $[0,1]$ sample over an infinite collection of intervals made up by successive positions of a multiplicative random walk independent of the uniform sample. We prove a law of the iterated logarithm for the number of non-empty (occupied) intervals as the size of the uniform sample becomes large.

math.PR↗
arXiv2016

Refined properties for the Mellin transform, with applications to convergence of families obtained by biasing or by the stationary excess operator

We first provide some properties of the Mellin transform of nonnegative random variables, such that monotonicity, injectivity and effect of size biasing. Convergence of Mellin transforms is also entirely formalized through convergence in distribution and uniform integrability. As an application, we study a problem raised by Harkness and Shantaram (1969) who obtained, under sufficient conditions, a limit theorem for sequences of nonnegative random variables build with the iterated stationary excess operator. We reformulate this problem through the concept of multiply monotone functions and through the convergence of families build by the continuous time version of the iterated stationary excess operator and also by size biasing. The latter allows us to show that in our context, continuous time convergence is equivalent to discrete time convergence, that the conditions of Harkness and Shantaram are actually necessary and that the only possible limits in distribution are mixture of exponential with lognormal distributions.

math.PR↗
arXiv2016

A generalization of the 2D Sleipian functions

The main content of this work is devoted to study various explicit family of special functions generalizing the famous 2D Sleipain functions, founded in 1960's by D. Slepian and his co-authors. As a consequence, many desirable spectral properties of the corresponding weighted finite Fourier transform are deduced from the rich literature. In particular, similar aspect related to Slepian's seminal papers is the investigation of differential operators that commute with appropriate integral operators are given. Finally, we provided the reader with some analytic expressions for the Fourier transforms of the Disk polynomials and the two variables Gegenbauer polynomials.

math.CA↗
arXiv2013

Intertwining operator associated to the complex Dunkl operator of type $G(m,1,N)$

In this work, we consider the Dunkl complex reflection operators related to the group $G(m,1,N)$ in the complex plane \begin{align*} T_i=\frac{\partial}{\partial z_i}+k_0\sum_{j\neq i}\sum_{r=0}^{m-1}\frac{1-s_i^{-r}(i,j)s_i^r} {z_i-\varepsilon^r z_j}+\sum_{j=1}^{m-1}k_j\sum_{r=0}^{m-1}\frac{\varepsilon^{-rj}s_i^r}{z_i}, \,\,1\leq i\leq N. \end{align*} We first review the theory of Dunkl operators for complex reflection groups we recall some results related to the hyper--Bessel functions, which are solutions of a higher order differential equation. Secondly, we construct a new explicit intertwining operator between the operator $T_i$ and the partial derivative operator $\frac{\partial}{\partial x_i}.$ As application we given an explicit solution of the system: $$ T_if(x)=κλ_i f(x),\,\,\, f(0)=1.$$

math.CA↗
arXiv2013

On the zeros of the big $q$-Bessel functions and applications

This paper deals with the study of the zeros of the big $q$-Bessel functions. In particular, we prove a new orthogonality relations for this functions similar to the one for the classical Bessel functions. Also we give some applications related to the sampling theory.

math.CV↗
arXiv2013

Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System

In this paper, we introduce a new differential-difference operator $T_ξ$ $(ξ\in \mathbb{R}^N)$ by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between $T_ξ$ and the directional derivative $\partial_ξ$. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator.

math.CA↗
arXiv2013

New addition formula for the little $q$-Bessel functions

Starting from the addition formula for little $q$-Jacobi polynomials, we derive a new addition formula for the little $q$-Bessel functions. The result is obtained by the use of a limit transition. We also establish a product formula for little $q$-Bessel functions with a positive and symmetric kernel.

math-ph↗
arXiv2013

$r$-extension of Dunkl operator in one variable and Bessel functions of vector index

In this work we present an operator $D_μ$ constructed with the help of the cyclic group set of the $r^{\small th}$ roots of unity. This operator constitute an $r$-extension of the Dunkl operator in one variable because when $r=2$ it reduces to the classical one and admits as eigenfunctions the Bessel functions of vector index early deeply studied by Klyuchantsev. This paper is argued by specific examples and contains some interesting results which are the prelude of harmonic analysis related to this operator.

math.FA↗