Linearization of Arbitrary products of classical orthogonal polynomials

Mahouton Hounkonnou; Said Belmehdi; André Ronveaux

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 2, page 187-196
  • ISSN: 1233-7234

Abstract

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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.

How to cite

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Hounkonnou, Mahouton, Belmehdi, Said, and Ronveaux, André. "Linearization of Arbitrary products of classical orthogonal polynomials." Applicationes Mathematicae 27.2 (2000): 187-196. <http://eudml.org/doc/219266>.

@article{Hounkonnou2000,
abstract = {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.},
author = {Hounkonnou, Mahouton, Belmehdi, Said, Ronveaux, André},
journal = {Applicationes Mathematicae},
keywords = {classical orthogonal polynomials; Hermite orthogonal polynomials; linearization coefficients; recurrence relations; differential equations},
language = {eng},
number = {2},
pages = {187-196},
title = {Linearization of Arbitrary products of classical orthogonal polynomials},
url = {http://eudml.org/doc/219266},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Hounkonnou, Mahouton
AU - Belmehdi, Said
AU - Ronveaux, André
TI - Linearization of Arbitrary products of classical orthogonal polynomials
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 2
SP - 187
EP - 196
AB - 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.
LA - eng
KW - classical orthogonal polynomials; Hermite orthogonal polynomials; linearization coefficients; recurrence relations; differential equations
UR - http://eudml.org/doc/219266
ER -

References

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