Publication: Orthogonal polynomials and cubic polynomial mappings. II. The positive-definite case
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2001
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Mesa State College
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Abstract
Let $\{P_n\}_{n\geq 0}$ be a sequence of polynomials orthogonal with respect to some distribution function $\sigma$ and let $\{Q_n\}_{n\geq 0}$ be a simple set (i.e., each $Q_n$ has degree exactly $n$) of polynomials such that $Q_{3n+m}(x)=\theta_m(x)P_n(\pi_3(x))$ for all $n=0,1,2,\dots$ where $\pi_3$ is a fixed monic polynomial of degree 3 and $\theta_m$ a fixed polynomial of degree $m$ with $0\leq m\leq 2$. We give necessary and sufficient conditions in order that $\{Q_n\}_{n\geq 0}$ be a sequence of polynomials orthogonal with respect to some distribution function $\tilde\sigma$. Under these conditions, we prove that $$d\tilde\sigma(x)=\sum _{i=1}M_i\delta_{x_i}(x)dx+\chi_{\pi_3 -1}([\xi,\eta])}(x)\left frac{\theta_{2-m}(x)}{\theta_m(x)}\right frac{d\sigma(\pi_3(x))}{\pi_3'(x)}$$ where $\chi_A$ means the characteristic function of the set $A$, $[\xi,\eta]$ is the support of $d\sigma$, $\theta_{2-m}$ denotes a polynomial of degree exactly $2-m$ and, if $m\geq 1$, $M_i$ is a mass located at the zero $x_i$ of $\theta_m(x)\equiv\prod _{i=1}(x-x_i),\ \delta_{x_i}(x)$ being the Dirac functional at the point $x_i$.
Description
10 pages, no figures.-- MSC1991 codes: Primary 42C05.
MR#: MR1862232 (2002g:42029)
MR#: MR1862232 (2002g:42029)
Keywords
Orthogonal polynomials, Polynomial mappings, Stieltjes transforms, Chain sequences
Bibliographic citation
Communications in the Analytic Theory of Continued Fractions, 2001, n. 9, p. 11-20