On differential equations for Sobolev-type Laguerre polynomials

Authors:
J. Koekoek, R. Koekoek and H. Bavinck

Journal:
Trans. Amer. Math. Soc. **350** (1998), 347-393

MSC (1991):
Primary 33C45; Secondary 34A35

DOI:
https://doi.org/10.1090/S0002-9947-98-01993-X

MathSciNet review:
1433121

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Abstract: The Sobolev-type Laguerre polynomials are orthogonal with respect to the inner product

where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form

where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.

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Additional Information

**J. Koekoek**

Affiliation:
Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600GA Delft, The Netherlands

**R. Koekoek**

Affiliation:
Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600GA Delft, The Netherlands

Email:
r.koekoek@twi.tudelft.nl

**H. Bavinck**

Affiliation:
Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600GA Delft, The Netherlands

Email:
h.bavinck@twi.tudelft.nl

DOI:
https://doi.org/10.1090/S0002-9947-98-01993-X

Keywords:
Differential equations,
Sobolev-type Laguerre polynomials

Received by editor(s):
August 28, 1995

Received by editor(s) in revised form:
June 24, 1996

Article copyright:
© Copyright 1998
American Mathematical Society