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Orthogonal polynomials and quadratic transformations

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URI: http://hdl.handle.net/10016/6263
ISSN: 0032-5155
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orthogonal_marcellan_pm_1999.pdf (232.84 KB)
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1999
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European Mathematical Society
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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].
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33 pages, no figures.-- MSC1991 code: Primary 42C05.
MR#: MR1680116 (2000b:42021)
Zbl#: Zbl 0936.42012
Keywords
Orthogonal polynomials, Recurrence coefficients, Polynomial mappings, Stieltjes functions
Bibliographic citation
Portugaliae Mathematica, 1999, vol. 56, n. 1, p. 81-113
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