Cubic decomposition of a Laguerre–Hahn linear functional I

Abstract

The aim of this contribution is to study orthogonal polynomials via cubic decomposition in the framework of the Laguerre–Hahn class. We consider two monic orthogonal polynomial sequences $\{W_n{\}}_{n\ge 0}$ and $\{P_n{\}}_{n\ge 0}$, and we let $w$ and $u$ be, respectively, the corresponding regular linear functionals such that $W_{3n}(x)=P_n(x^3)$, $n\ge 0$. We prove that if either $w$ or $u$ is a Laguerre–Hahn linear functional, then so is the other one. Based on this result, we deduce a complete analysis of the class $s$ of the Laguerre–Hahn linear functional $w$. More precisely, we show that $3s^{\prime }\le s\le 3s^{\prime }+6$, where $s^{\prime }$ is the class of $u$. An illustrative example of class $1$ is analyzed.