Linearization of Arbitrary products of classical orthogonal polynomials

Volume 27 / 2000

Mahouton Hounkonnou, Said Belmehdi, André Ronveaux Applicationes Mathematicae 27 (2000), 187-196 DOI: 10.4064/am-27-2-187-196

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.

Authors

  • Mahouton Hounkonnou
  • Said Belmehdi
  • André Ronveaux

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