Second degree classical forms |
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Authors: | Driss Beghdadi Pascal Maroni |
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Institution: | Université Pierre et Marie Curie, Laboratoire d'Analyse Numérique, 4 Place Jussieu, 75252 Paris Cedex 05, France |
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Abstract: | A form (linear functional) u is called regular if there exists a sequence of polynomials {Pn}n≥0, deg Pn = n which is orthogonal with respect to u. Such a form is said to be of second degree if there are polynomials B and C such that the Stieltjes function satisfies a relation of the form BS2(u) + CS(u) + D = 0.Classical forms correspond to classical orthogonal polynomials: sequences of polynomials whose derivatives also form an orthogonal sequence. In this paper, the authors determine all the classical forms which are of second degree. They show that Hermite, Laguerre and Bessel forms are not of second degree. Only Jacobi forms which satisfy a certain condition possess this property. |
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