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# Generating Some Symmetric Semi-classical Orthogonal Polynomials

Published:2015-07-20
Printed: Dec 2015
• Mohamed Zaatra,
Institut Supérieur des Sciences et Techniques des Eaux de Gabès, Campus universitaire, Gabès 6072, Tunisia
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## Abstract

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
 MSC Classifications: 33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

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