arXiv subjects

Mohamed Khalfallah

Publications and source records attributed to Mohamed Khalfallah.

3 recordsLinked to original sources

arXiv2026

On the moments of Laguerre-Hahn linear forms of class zero

The purpose of this paper is to provide a constructive method for generating the moments of Laguerre-Hahn linear forms of class zero. The approach relies on a general algebraic system on the moments, equivalent to the Laguerre-Hahn functional equation for forms of arbitrary class, from which the restriction to class zero yields explicit recurrences allowing the entire moment sequence to be built recursively from the first moment. These recurrences are applied to each of the ten canonical class-zero families, and the corresponding first moments are derived explicitly via a {\it Mathematica$^{\circledR}$} implementation.

math.GM↗

arXiv2026

A description of the Laguerre-Hahn orthogonal polynomials of class zero revisited

This paper has a threefold aim. On the one hand, we provide a complete description of Laguerre-Hahn forms of class zero. This fills a gap in the literature: more precisely, up to an affine change of variables, there are ten families, including two new ones analogous to the Bessel classical family. On the other hand, we establish the structure relations for all these families, correcting those that have previously been reported in the literature. At last, as an application, using an algorithm previously obtained, with the aid of symbolic computations, we derive four new structure relations and a new fourth-order differential equation for one of the new families analogous to Bessel.

math.CA↗

arXiv2026

On a general method for deriving a fourth-order differential equation satisfied by Laguerre-Hahn orthogonal polynomials with new results for the class 0 analogous to Hermite

In this work, we develop a constructive method for deriving four structure relations and a fourth-order linear differential equation satisfied by Laguerre-Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre-Hahn polynomials of class zero, analogous to the Hermite case. In addition, we present results for a semiclassical example of class 1.

math.CA↗