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Generating Some Symmetric Semi-classical Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

Mohamed Zaatra
Mohamed Zaatra*
Affiliation:
Institut Supérieur des Sciences et Techniques des Eaux de Gabès, Campus universitaire, Gabès 6072, Tunisia e-mail: medzaatra@yahoo.fr
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Abstract

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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 $\left( \sigma f \right)\left( x \right)\,=\,f\left( {{x}^{2}} \right)$ 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

33C4542C05orthogonal polynomialsquadratic decompositionsemi-classical formsstructure relation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 877 - 890
Copyright
Copyright © Canadian Mathematical Society 2015

References

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