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The Krall Polynomials as Solutions to a Second Order Differential Equation

Published online by Cambridge University Press:  20 November 2018

Lance L. Littlejohn*
Affiliation:
Department of Mathematics, Computer Science, and Systems Design, The University of Texasat San Antonio, 78285
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Abstract

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A popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form

1.1

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Atkinson, F. V. and Everitt, W. N., Orthogonal Polynomials Which Satisfy Second Order Differential Equations, Christoffel Festschrift, 1981, Birhaüser, Basel.Google Scholar
2. Hahn, W., On Differential Equations for Orthogonal Polynomials, Funkcialaj Ekvacioj, 21 (1978) 1-9.Google Scholar
3. Hahn, W., Über Orthogonalpolynome mit drei Parametern, Deutsche Math., 5 (1940) 273-278.Google Scholar
4. Koornwinder, T. H., A Further Generalization of KralVs Jacobi type Polynomials, Mathematisch Centrum, Amsterdam, 1982.Google Scholar
5. Krall, A. M., Orthogonal Polynomials Satisfying Fourth Order Differential Equations, Proc. Roy. Soc. Edin., 87(A) (1981) 271-288.Google Scholar
6. Krall, H. L., Certain Differential Equations for Tchebycheff Polynomials, Duke Math. Jour., 4 (1938) 705-718.Google Scholar
7. Krall, H. L., On Orthogonal Polynomials Satisfying a Certain Fourth Order Differential Equation, The Pennsylvania State College Studies, No. 6, The Pennsylvania State College, State College, Pa., 1940.Google Scholar
8. Littlejohn, L. L., The Krall Polynomials: A new Class of Orthogonal Polynomials, Quaestione Mathematicae, 5 (1982), 255-265.Google Scholar
9. Littlejohn, L. L. and Shore, S. D., Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations, Can. Math. Bull., 25 (3) 1982.Google Scholar
10. Shore, S. D., On the Second Order Differential Equation which has Orthogonal Polynomial Solutions, Bull. Calcutta Math. Soc, 56 (1964) 195-198.Google Scholar