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Linearization of the product of orthogonal polynomials of a discrete variable

100%

Belmehdi S., Lewanowicz S., Ronveaux A.

Applicationes Mathematicae

1996-1997

tom 24

nr 4

445-455

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 $P_iP_j=\sum_kc(i,j,k)P_k$, 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 {$P_k$}.

Linearization of Arbitrary products of classical orthogonal polynomials

100%

Hounkonnou M. N., Belmehdi S., Ronveaux A.

Applicationes Mathematicae

2000

tom 27

nr 2

187-196

A procedure is proposed in order to expand $w=\prod^N_{j=1} P_{i_j}(x)=\sum^M_{k=0} L_ k P_ k(x)$ where $P_i(x)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ($M=\sum^N_{j=1} i_j$). 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.

Ograniczanie wyników

2 Applicationes Mathematicae

2 Belmehdi S.

2 Ronveaux A.

1 Hounkonnou M. N.

1 Lewanowicz S.

1 2000

1 1997

Wyniki wyszukiwania

Linearization of the product of orthogonal polynomials of a discrete variable

Linearization of Arbitrary products of classical orthogonal polynomials