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Search: All articles in the CMB digital archive with keyword orthogonal polynomials

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1. CMB Online first

Zaatra, Mohamed
Generating some symmetric semi-classical orthogonal polynomials
We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relation
Categories:33C45, 42C05

2. CMB 2001 (vol 44 pp. 337)

Vinet, Luc; Zhedanov, Alexei
Spectral Transformations of the Laurent Biorthogonal Polynomials, II. Pastro Polynomials
We continue to study the simplest closure conditions for chains of spectral transformations of the Laurent biorthogonal polynomials ($\LBP$). It is shown that the 1-1-periodic $q$-closure condition leads to the $\LBP$ introduced by Pastro. We introduce classes of semi-classical and Laguerre-Hahn $\LBP$ associated to generic closure conditions of the chain of spectral transformations.

Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformations

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