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Linearization of Arbitrary products of classical orthogonal polynomials

Mahouton HounkonnouSaid BelmehdiAndré Ronveaux — 2000

Applicationes Mathematicae

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

Linearization of the product of orthogonal polynomials of a discrete variable

Saïd BelmehdiStanisław LewanowiczAndré Ronveaux — 1997

Applicationes Mathematicae

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 ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by P k .

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