Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T04:33:53.811Z Has data issue: false hasContentIssue false
  • English
  • Français

The Krall Polynomials as Solutions to a Second Order Differential Equation

Published online by Cambridge University Press:  20 November 2018

Lance L. Littlejohn
Lance L. Littlejohn*
Affiliation:
Department of Mathematics, Computer Science, and Systems Design, The University of Texasat San Antonio, 78285
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

33A65
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 4 , 01 December 1983 , pp. 410 - 417
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