Saïd Esteban Belmehdi - Academia.edu (original) (raw)

Papers by Saïd Esteban Belmehdi

Research paper thumbnail of Forme lineaires et polynomes orthogonaux semi-classiques de classe s=1. Description et classification

Http Www Theses Fr, 1990

Les suites des polynomes orthogonaux semi-classiques de classe s sont des suites orthogonales don... more Les suites des polynomes orthogonaux semi-classiques de classe s sont des suites orthogonales dont la suite des derivees est quasi-orthogonale d'ordre s, elles constituent une extension naturelle des suites orthogonales classiques, qui sont des suites orthogonales dont la suite des derivees est aussi orthogonale. Dans ce memoire, apres l'exposition des notions generales sur l'orthogonalite et la quasi-orthogonalite on empreinte deux voies pour explorer les polynomes semi-classiques de classe s=1; la voie directe, a savoir, construire des nouvelles suites semi-classiques a partir des suites semi-classiques connues via des modifications des formes lineaires, ainsi en appliquant ce procede aux suites classiques (hermite, laguerre, bessel, jacobi) on obtient des suites orthogonales semi-classiques de classe s=1 parfaitement determinees. La deuxieme voie consiste a repertorier les equations fonctionnelles de classe s=1, etablir les equations canoniques et les resoudre via differentes techniques, entre autres, on donne des representations integrales de forme semi-classique de classe s=1. Ce memoire se termine sur le systeme non lineaire verifie par les coefficients de la relation de recurrence d'ordre deux. Orthogonalite, quasi-orthogonalite, relation de recurrence d'ordre deux, suites orthogonales classiques et semi-classiques, modifications des formes lineaires, equations differentielles du 2eme ordre du type sturm-liouville et du type laguerre-perron, relation de structure, equations fonctionnelles canoniques des formes lineaires semi-classiques de classe s=1, formes lineaires classiques tronquees, systeme non lineaire verifie par les coefficients de la relation de recurrence d'ordre deux

Research paper thumbnail of On the left multiplication of a regular linear functional by a polynomial

Research paper thumbnail of Another characterization of the semi-classical orthogonal polynomials

Research paper thumbnail of Interlacing properties of zeros for the derivative, associated and adjacent of orthogonal polynomial, semi-classical and classical cases

... For instance, tn;i = xn?1;i for Hermite polynomials Hn(x), (Appel property) and zn;i = tn;i f... more ... For instance, tn;i = xn?1;i for Hermite polynomials Hn(x), (Appel property) and zn;i = tn;i for Chebychev polynomials Tn(x). The aim of this work ... for which it is known that zn;i = tn;i. In the Hermite case (K = 2) and in the Gegenbauer case (K = 1+2 ), Laforgia and Elbert have proved ...

Research paper thumbnail of On Appell-Laguerre polynomials

Journal of Computational and Applied Mathematics, 1993

In this note we give a digest study of Appell-Laguerre polynomials, we provide a recurrence relat... more In this note we give a digest study of Appell-Laguerre polynomials, we provide a recurrence relation and a second-order differential equation satisfied by these polynomials. Moreover, an explicit expression and a generating function of the polynomials are given. The location and the number of the real zeros of Appell-Laguerre polynomials are determined; consequently, we recover information about the zeros of Laguerre polynomials when the range of the parameter (Y is the whole real axis excepting the negative integers.

Research paper thumbnail of On the associated orthogonal polynomials

Journal of Computational and Applied Mathematics, 1990

By using the second-order recurrence relation this paper gives some new results on associated ort... more By using the second-order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions' tool. Some results are very useful for obtaining the second-order differential equation satisfied by the semi-classical orthogonal polynomials (Hendriksen and van Rossum (1985) Maroni (1987)) (cf. Section 3). Also, the main formula derives from Proposition 2.6, by which the fourth-order differential equation, satisfied by some Laguerre-Hahn polynomials (Magnus (1984)), is obtained (cf. Behnehdi and Ronveaux (1989), Dini et al. (1989), Ronveaux et al. (1990)).

Research paper thumbnail of Integral representation of the solutions to Heun's biconfluent equation

Abstract and Applied Analysis, 2004

First, we trace the genesis of the canonical form of Heun's biconfluent equation. Second, we ... more First, we trace the genesis of the canonical form of Heun's biconfluent equation. Second, we present a method which allows us to find an integral expression as a solution to our equation, and finally, using the properties of MeijerG-functions, we give an integral representation of a fundamental system of solutions to the biconfluent equation.

Research paper thumbnail of On semi-classical linear functionals of class s=1. Classification and integral representations

Indagationes Mathematicae, 1992

In recent years several papers have been published in quantum mechanical computation, kinetic the... more In recent years several papers have been published in quantum mechanical computation, kinetic theory, statistical applications and so forth, dealing with the so-called non-classical orthogonal

Research paper thumbnail of Fourth-order di erential equation satis ed by the associated orthogonal polynomials

Research paper thumbnail of Linearization of products of orthogonal polynomials of a discrete variable

Applicationes Mathematicae

Research paper thumbnail of Fourth-order differential equation satis ed by the associated orthogonal polynomials

Research paper thumbnail of Laguerre-Freud equations for the recurrence coe cients of semi-classical orthogonal polynomials

Journal of Approximation Theory, 1994

Research paper thumbnail of Generalized linearization problems

Journal of Physics A: Mathematical and General, 1995

We prove that if Pi(x) and Pj(x) are two families of semi-classical orthogonal polynomials, all t... more We prove that if Pi(x) and Pj(x) are two families of semi-classical orthogonal polynomials, all the linearization coefficients Li,j,k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index. This property also extends to the linearization coefficients arising from an arbitrary number of products of semi-classical orthogonal polynomials.

Research paper thumbnail of About polynomials related to multiphotonic bremsstrahlung effects

Journal of Physics A: Mathematical and General, 1991

ABSTRACT

Research paper thumbnail of Fourth-order differential equation for the co-modified of semi-classical orthogonal polynomials

Journal of Computational and Applied Mathematics, 1990

We give a new derivation of the fourth-order differential equation satisfied by the co-modified (... more We give a new derivation of the fourth-order differential equation satisfied by the co-modified (orthogonal polynomials) of any semi-classical family of orthogonal polynomials. This procedure generalizes a technique already used in the co-recursive and co-dilated cases. For the classical cases we give explicitly the corresponding differential equation in a determinantal form.

Research paper thumbnail of Orthogonality with respect to the sum of two semiclassical regular linear forms

Journal of Computational and Applied Mathematics, 1991

Ronveaux, A., S. Belmehdi and F. Marcellkr, Orthogonality with respect to the sum of two semiclas... more Ronveaux, A., S. Belmehdi and F. Marcellkr, Orthogonality with respect to the sum of two semiclassical regular linear forms, Journal of Computational and Applied Mathematics 37 (1991) 2655272. The linear form sum of two semiclassical regular linear forms verifies in general a second-order differential equation and we examine some situations where this new form remains semiclassical. The sequences of polynomials orthogonal with respect to this new form, called of second category, have associated polynomials which do not belong to the Laguerre-Hahn class.

Research paper thumbnail of Laguerre-Freud′s Equations for the Recurrence Coefficients of Semi-classical Orthogonal Polynomials

Journal of Approximation Theory, 1994

Research paper thumbnail of Orthogonal polynomials associated with some modifications of a linear functional

Applicable Analysis, 1992

ABSTRACT

Research paper thumbnail of Linearization of the product of orthogonal polynomials of a discrete variable

Applicationes Mathematicae, 1997

Let {P k } be any sequence of classical orthogonal polynomials of a discrete variable. We give ex... more Let {P k } be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in P i P j = k c(i, j, k)P k , in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function , and the coefficients of the three-term recurrence relation and of two structure relations obeyed by {P k }.

Research paper thumbnail of Generalized Gegenbauer orthogonal polynomials

Journal of Computational and Applied Mathematics, 2001

In this paper we explore a speciÿc semi-classical orthogonal sequence, namely the generalized Geg... more In this paper we explore a speciÿc semi-classical orthogonal sequence, namely the generalized Gegenbauer orthogonal polynomials (GG) which appear in many applications such as the weighted L p mean convergence of Hermite-Fejà er interpolation or the chain of harmonic oscillators in the absence of externally applied forces. First we trace back the genesis of GG underlining its links with the Jacobi orthogonal polynomials. Second we establish a di erential-di erence relation and the second-order di erential equation satisÿed by this sequence. We end by giving the fourth-order di erential equation satisÿed by the association (of arbitrary order) of the GG.

Research paper thumbnail of Forme lineaires et polynomes orthogonaux semi-classiques de classe s=1. Description et classification

Http Www Theses Fr, 1990

Les suites des polynomes orthogonaux semi-classiques de classe s sont des suites orthogonales don... more Les suites des polynomes orthogonaux semi-classiques de classe s sont des suites orthogonales dont la suite des derivees est quasi-orthogonale d'ordre s, elles constituent une extension naturelle des suites orthogonales classiques, qui sont des suites orthogonales dont la suite des derivees est aussi orthogonale. Dans ce memoire, apres l'exposition des notions generales sur l'orthogonalite et la quasi-orthogonalite on empreinte deux voies pour explorer les polynomes semi-classiques de classe s=1; la voie directe, a savoir, construire des nouvelles suites semi-classiques a partir des suites semi-classiques connues via des modifications des formes lineaires, ainsi en appliquant ce procede aux suites classiques (hermite, laguerre, bessel, jacobi) on obtient des suites orthogonales semi-classiques de classe s=1 parfaitement determinees. La deuxieme voie consiste a repertorier les equations fonctionnelles de classe s=1, etablir les equations canoniques et les resoudre via differentes techniques, entre autres, on donne des representations integrales de forme semi-classique de classe s=1. Ce memoire se termine sur le systeme non lineaire verifie par les coefficients de la relation de recurrence d'ordre deux. Orthogonalite, quasi-orthogonalite, relation de recurrence d'ordre deux, suites orthogonales classiques et semi-classiques, modifications des formes lineaires, equations differentielles du 2eme ordre du type sturm-liouville et du type laguerre-perron, relation de structure, equations fonctionnelles canoniques des formes lineaires semi-classiques de classe s=1, formes lineaires classiques tronquees, systeme non lineaire verifie par les coefficients de la relation de recurrence d'ordre deux

Research paper thumbnail of On the left multiplication of a regular linear functional by a polynomial

Research paper thumbnail of Another characterization of the semi-classical orthogonal polynomials

Research paper thumbnail of Interlacing properties of zeros for the derivative, associated and adjacent of orthogonal polynomial, semi-classical and classical cases

... For instance, tn;i = xn?1;i for Hermite polynomials Hn(x), (Appel property) and zn;i = tn;i f... more ... For instance, tn;i = xn?1;i for Hermite polynomials Hn(x), (Appel property) and zn;i = tn;i for Chebychev polynomials Tn(x). The aim of this work ... for which it is known that zn;i = tn;i. In the Hermite case (K = 2) and in the Gegenbauer case (K = 1+2 ), Laforgia and Elbert have proved ...

Research paper thumbnail of On Appell-Laguerre polynomials

Journal of Computational and Applied Mathematics, 1993

In this note we give a digest study of Appell-Laguerre polynomials, we provide a recurrence relat... more In this note we give a digest study of Appell-Laguerre polynomials, we provide a recurrence relation and a second-order differential equation satisfied by these polynomials. Moreover, an explicit expression and a generating function of the polynomials are given. The location and the number of the real zeros of Appell-Laguerre polynomials are determined; consequently, we recover information about the zeros of Laguerre polynomials when the range of the parameter (Y is the whole real axis excepting the negative integers.

Research paper thumbnail of On the associated orthogonal polynomials

Journal of Computational and Applied Mathematics, 1990

By using the second-order recurrence relation this paper gives some new results on associated ort... more By using the second-order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions' tool. Some results are very useful for obtaining the second-order differential equation satisfied by the semi-classical orthogonal polynomials (Hendriksen and van Rossum (1985) Maroni (1987)) (cf. Section 3). Also, the main formula derives from Proposition 2.6, by which the fourth-order differential equation, satisfied by some Laguerre-Hahn polynomials (Magnus (1984)), is obtained (cf. Behnehdi and Ronveaux (1989), Dini et al. (1989), Ronveaux et al. (1990)).

Research paper thumbnail of Integral representation of the solutions to Heun's biconfluent equation

Abstract and Applied Analysis, 2004

First, we trace the genesis of the canonical form of Heun's biconfluent equation. Second, we ... more First, we trace the genesis of the canonical form of Heun's biconfluent equation. Second, we present a method which allows us to find an integral expression as a solution to our equation, and finally, using the properties of MeijerG-functions, we give an integral representation of a fundamental system of solutions to the biconfluent equation.

Research paper thumbnail of On semi-classical linear functionals of class s=1. Classification and integral representations

Indagationes Mathematicae, 1992

In recent years several papers have been published in quantum mechanical computation, kinetic the... more In recent years several papers have been published in quantum mechanical computation, kinetic theory, statistical applications and so forth, dealing with the so-called non-classical orthogonal

Research paper thumbnail of Fourth-order di erential equation satis ed by the associated orthogonal polynomials

Research paper thumbnail of Linearization of products of orthogonal polynomials of a discrete variable

Applicationes Mathematicae

Research paper thumbnail of Fourth-order differential equation satis ed by the associated orthogonal polynomials

Research paper thumbnail of Laguerre-Freud equations for the recurrence coe cients of semi-classical orthogonal polynomials

Journal of Approximation Theory, 1994

Research paper thumbnail of Generalized linearization problems

Journal of Physics A: Mathematical and General, 1995

We prove that if Pi(x) and Pj(x) are two families of semi-classical orthogonal polynomials, all t... more We prove that if Pi(x) and Pj(x) are two families of semi-classical orthogonal polynomials, all the linearization coefficients Li,j,k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index. This property also extends to the linearization coefficients arising from an arbitrary number of products of semi-classical orthogonal polynomials.

Research paper thumbnail of About polynomials related to multiphotonic bremsstrahlung effects

Journal of Physics A: Mathematical and General, 1991

ABSTRACT

Research paper thumbnail of Fourth-order differential equation for the co-modified of semi-classical orthogonal polynomials

Journal of Computational and Applied Mathematics, 1990

We give a new derivation of the fourth-order differential equation satisfied by the co-modified (... more We give a new derivation of the fourth-order differential equation satisfied by the co-modified (orthogonal polynomials) of any semi-classical family of orthogonal polynomials. This procedure generalizes a technique already used in the co-recursive and co-dilated cases. For the classical cases we give explicitly the corresponding differential equation in a determinantal form.

Research paper thumbnail of Orthogonality with respect to the sum of two semiclassical regular linear forms

Journal of Computational and Applied Mathematics, 1991

Ronveaux, A., S. Belmehdi and F. Marcellkr, Orthogonality with respect to the sum of two semiclas... more Ronveaux, A., S. Belmehdi and F. Marcellkr, Orthogonality with respect to the sum of two semiclassical regular linear forms, Journal of Computational and Applied Mathematics 37 (1991) 2655272. The linear form sum of two semiclassical regular linear forms verifies in general a second-order differential equation and we examine some situations where this new form remains semiclassical. The sequences of polynomials orthogonal with respect to this new form, called of second category, have associated polynomials which do not belong to the Laguerre-Hahn class.

Research paper thumbnail of Laguerre-Freud′s Equations for the Recurrence Coefficients of Semi-classical Orthogonal Polynomials

Journal of Approximation Theory, 1994

Research paper thumbnail of Orthogonal polynomials associated with some modifications of a linear functional

Applicable Analysis, 1992

ABSTRACT

Research paper thumbnail of Linearization of the product of orthogonal polynomials of a discrete variable

Applicationes Mathematicae, 1997

Let {P k } be any sequence of classical orthogonal polynomials of a discrete variable. We give ex... more Let {P k } be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in P i P j = k c(i, j, k)P k , in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function , and the coefficients of the three-term recurrence relation and of two structure relations obeyed by {P k }.

Research paper thumbnail of Generalized Gegenbauer orthogonal polynomials

Journal of Computational and Applied Mathematics, 2001

In this paper we explore a speciÿc semi-classical orthogonal sequence, namely the generalized Geg... more In this paper we explore a speciÿc semi-classical orthogonal sequence, namely the generalized Gegenbauer orthogonal polynomials (GG) which appear in many applications such as the weighted L p mean convergence of Hermite-Fejà er interpolation or the chain of harmonic oscillators in the absence of externally applied forces. First we trace back the genesis of GG underlining its links with the Jacobi orthogonal polynomials. Second we establish a di erential-di erence relation and the second-order di erential equation satisÿed by this sequence. We end by giving the fourth-order di erential equation satisÿed by the association (of arbitrary order) of the GG.