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ArticleOriginal scientific text

Title

Linearization of Arbitrary products of classical orthogonal polynomials

Authors 1, 2, 3

Affiliations

  1. Unité de Recherche en Physique Théorique (URPT), Institut de Mathématique et de Sciences Physiques (IMSP), Université Nationale du Benin, BP 613 Porto-Novo, Benin
  2. UFR de Mathématique, Université des Sciences et Technologies, Lille I, F-59655 Villeneuve d'Ascq, France
  3. Facultés Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium

Abstract

A procedure is proposed in order to expand w=∏N_{j=1}Pij(x)=∑M_{k=0}LkPk(x) where Pi(x) belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) (M=∑N_{j=1}ij). We first derive a linear differential equation of order 2N satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients Lk. We develop in detail the two cases [Pi(x)]N, Pi(x)Pj(x)Pk(x) and give the recurrencerelation in some cases (N=3,4), when the polynomials Pi(x)are monic Hermite orthogonal polynomials.

Keywords

ENG
classical orthogonal polynomials, Hermite orthogonal polynomials, linearization coefficients, recurrence relations, differential equations

Bibliography

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Applicationes Mathematicae
2000/27/2
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Pages:
187-196
Main language of publication
English
Received
1998-09-30
Accepted
1999-05-18
Published
2000
Exact and natural sciences