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

#### Abstract

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.

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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