Orthogonal Polynomials and Quadratic Transformations

Francisco Marcellán and José Petronilho

Abstract: Starting from a sequence $\{P_n\}_{n\geq 0}$ of monic polynomials orthogonal with respect to a linear functional ${\bf u}$, we find a linear functional ${\bf v}$ such that $\{Q_n\}_{\geq 0}$, with either $Q_{2n}(x)=P_n(T(x))$ or $Q_{2n+1}(x)=(x-a)\,P_n(T(x))$ where $T$ is a monic quadratic polynomial and $a\in\C$, is a sequence of monic orthogonal polynomials with respect to ${\bf v}$. In particular, we discuss the case when ${\bf u}$ and ${\bf v}$ are both positive definite linear functionals. Thus, we obtain a solution for an inverse problem which is a converse, for quadratic mappings, of one analyzed in [11].

Keywords: Orthogonal polynomials; recurrence coefficients; polynomial mappings; Stieltjes functions.

Classification (MSC2000): 42C05.

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