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Generating Some Symmetric Semi-classical Orthogonal Polynomials
Published online by Cambridge University Press: 20 November 2018
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 $\left( \sigma f \right)\left( x \right)\,=\,f\left( {{x}^{2}} \right)$ 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.
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- Copyright © Canadian Mathematical Society 2015