This article is cited in 3 papers

Connection formulas and representations of Laguerre polynomials in terms of the action of linear differential operators

B. Aloui^a, L. Khériji<sup>b

a</sup> Université de Gabès, Institut Supérieur des Systèmes Industriels de Gabès, Rue Salah Eddine Elayoubi 6033 Gabès, Tunisia
^b Université de Tunis El Manar, Institut Préparatoire aux Etudes d’Ingénieur El Manar, Campus Universitaire El Manar, B.P. 244, 2092 Tunis, Tunisia

Abstract: In this paper, we introduce the notion of $\mathfrak{O}_{\varepsilon}$-classical orthogonal polynomials, where $\mathfrak{O}_{\varepsilon}:=\mathbb{I}+\varepsilon D$ ($\varepsilon\neq0$). It is shown that the scaled Laguerre polynomial sequence $\{a^{-n}L^{(\alpha)}_n(ax)\}_{n\geq0}$, where $a=-\varepsilon^{-1}$, is actually the only $\mathfrak{O}_{\varepsilon}$-classical sequence. As an illustration, we deal with some representations of Laguerre polynomials $L^{(0)}_n(x)$ in terms of the action of linear differential operators on the Laguerre polynomials $L^{(m)}_n(x)$. The inverse connection problem of expanding Laguerre polynomials $L^{(m)}_n(x)$ in terms of $L^{(0)}_n(x)$ is also considered. Furthermore, some connection formulas between the monomial basis $\{x^n\}_{n\geq0}$ and the shifted Laguerre basis $\{L^{(m)}_n(x+1)\}_{n\geq0}$ are deduced.

Keywords: classical polynomials, Laguerre polynomials, lowering and raising operators, structure relations, higher order differential operators, connection formulas.

UDC: 517.587, 517.521.1

MSC: 33C45, 42C05

Received: 14.05.2019
Revised: 01.10.2019
Accepted: 23.09.2019

Language: English

DOI: 10.15393/j3.art.2019.6290

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