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Orthogonal Polynomials With Weight Function (1 - x)α( l + x)β + Mδ(x + 1) + Nδ(x - 1)

Published online by Cambridge University Press:  20 November 2018

Tom H. Koornwinder
Tom H. Koornwinder*
Affiliation:
Mathemattsch CentrumP.O. Box 4079, 1009 AB Amsterdam, Netherlands
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Abstract

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We study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

Keywords

33A6533A30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 205 - 214
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Atkinson, F. V. and Everitt, W. N., Orthogonal polynomials which satisfy second order differential equations, in “E. B. Christoffel, the influence of his work on mathematics and the physical sciences” (Butzer, P. L. and Fehér, F., eds.), Birkhäuser, 1981, pp. 173–181.Google Scholar
2. Erdélyi, A., e.a., , Higher transcendental functions, Vol. I, McGraw-Hill, 1953.Google Scholar
3. Erdélyi, A., e.a., , Higher transcendental functions, Vol. II, McGraw-Hill, 1953.Google Scholar
4. Hahn, W., Über Orthogonalpolynome mit besonderen Eigenschaften, in “E. B. Christoffel, the influence of his work on mathematics and the physical sciences” (Butzer, P. L. and Fehér, F., eds.), Birkhäuser, 1981, pp. 182–189.Google Scholar
5. Krall, A. M., Orthogonal polynomials satisfying fourth order differential equations, Proc. Royal. Soc. Edinburgh 87A (1981), 271–288.Google Scholar
6. Krall, H. L., Certain differential equations for Tchebycheff polynomials. Duke Math. J. 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, 1940.Google Scholar
8. Littlejohn, L. L., The Krall polynomials: A new class of orthogonal polynomials, Quaestiones Math. 5 (1982), 255–265.Google Scholar
9. Littlejohn, L. L. and Shore, S. D., Nonclassical orthogonal polynomials as solutions to second order differential equations, Canad. Math. Bull. 25 (1982), 291–295.Google Scholar
10. Niblett, J. D., Some hypergeometric identities, Pacific J. Math. 2 (1952), 219–225.Google Scholar
11. Nikishin, E. M., The Fade approximants, Proceedings International Congress of Mathematicians Helsinki 1978, Vol. II, pp. 623–630, Helsinki, 1980.Google Scholar
12. Szegö, G., Orthogonal polynomials, American Mathematical Society, Fourth edition, 1975.Google Scholar