Orthogonal Polynomials Associated with an Inverse Spectral Transform. The Cubic Case

Mabrouk Sghaier, Lamaa Khaled


The purpose of this work is to give  some new algebraic proprieties of the orthogonality of a
monic polynomial sequence $\{Q_n\}_{n\geq0}$ defined   by $$ Q_n(x)=P_n(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x) +r_nP_{n-3}(x),\quad n\geq1, $$
where $r_n \neq 0, n \geq 3$ and  $\{P_n\}_{n\geq0}$ is a given sequence of monic orthogonal polynomials. Essentially, we   consider some cases in which the parameters $ r_n $, $ s_n $, and $ t_n $ can be
computed more easily. Also, as a consequence, a matrix interpretation using
$LU$ and $UL$ factorization is done.  Some applications for Laguerre
and Tchebychev orthogonal polynomials of second kind are obtained.

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