A family of singular semi-classical functionals. (English) Zbl 1031.42025

Medem, J. C.

Indag. Math., New Ser. 13, No. 3, 351-362 (2002).

The distributional equation of a semi-classical functional provides an efficient way to study the properties of the semi-classical OPS/functionals. A particular case of semi-classical OPS/functionals are the classical ones. For the distributional equation of classical functionals a regularity condition holds. The main result of the present paper is to show that in general such condition does not hold. The examples of semi-classical functionals that satisfy certains singular equation are built and an explicit representation of the corresponding monic orthogonal polynomials as well as the recurrence relations are found.

Reviewer: Leonid Golinskii (Kharkov)

MSC:

42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33E30	Other functions coming from differential, difference and integral equations

Keywords:

semi-classical functionals; distributional equation; quasi-orthogonality; regularity condition; admissibility condition; orthogonal polynomials

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