Quadratic decomposition of a Laguerre-Hahn polynomial sequence I

Abstract

Given two sequences of monic orthogonal polynomials $\{P_{n}\}_{_{n\geq 0}}$ and $\{B_{n}\}_{_{n\geq 0}}$ such that $B_{2n}(x)=P_{n}(x^{2}),n\geq0,$ we show that the Laguerre-Hahn character of one of them remains valid for the other. Then we give relations between their classes and the coefficients of their structure relations. As an application, with an appropriate choice of the sequence $\{P_{n}\}_{n\geq 0},$ we obtain a new nonsymmetric semi-classical sequence of polynomials $\{B_{n}\}_{_{n\geq 0}}$ of class $s=1$.