Generating Some Symmetric Semi-classical Orthogonal Polynomials

Zaatra, Mohamed

Canadian mathematical bulletin, Tome 58 (2015) no. 4, pp. 877-890

Résumé

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 $v$ . Weconclude with an illustrative example.

Détail

DOI : 10.4153/CMB-2015-041-9

Mots-clés : 33C45, 42C05, orthogonal polynomials, quadratic decomposition, semi-classical forms, structurerelation