Proceedings of the American Mathematical Society

Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

Author(s): H. Bavinck
Journal: Proc. Amer. Math. Soc. 125 (1997), 3561-3567.
MSC (1991): Primary 33C45, 34A35
Retrieve article in: PDF
This article is available free of charge

Abstract: We consider the polynomials $\left\{ L_n^{\alpha ,M}(x,l)\right\} _{n=0}^\infty$ orthogonal with respect to the Sobolev type inner product

$\begin{equation*}\left\langle p,q\right\rangle =\frac 1{\Gamma (\alpha +1)}\int _0^\infty p(x)q(x)x^\alpha e^{-x}dx+Mp^{(l)}(0)q^{(l)}(0), \end{equation*}$

where $\alpha >-1,M\geq 0$ and $l$

is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order $2\alpha +4l+4$ if $\alpha$ is a nonnegative integer and $M>0.$

1.: H. Bavinck: A direct approach to Koekoek's differential equation for generalized Laguerre polynomials. Acta Math. Hungar. 66 (3) (1995), 247-253.MR 96a:33009
2.: H. Bavinck: Linear perturbations of differential or difference operators with polynomials as eigenfunctions. J. Comp. Appl. Math. 78 (1997), 179-195.
3.: H. Bavinck: A new result for Laguerre polynomials. J. Physics A. 29 (1996), L277-L279. CMP 96:15
4.: I.H. Jung, K.H. Kwon and G.J. Yoon: Differential equations of infinite order for Sobolev-type orthogonal polynomials. J. Comp. Appl. Math. 78 (1997), 277-293.
5.: A.M. Krall: Orthogonal polynomials satisfying fourth order differential equations. Proc. Roy. Soc. Edinburgh, 87A (1981), 271-288.MR 82d:33021
6.: A.M. Krall and L.L. Littlejohn: On the classification of differential equations having orthogonal polynomial solutions II. Ann. Mat. Pura Appl. (4) 149 (1987), 77-102. MR 89e:34017
7.: H.L. Krall: 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.
8.: L.L. Littlejohn: An application of a new theorem on orthogonal polynomials and differential equations. Quaestiones Math. 10 (1986), 49-61.MR 87m:42021
9.: J. Koekoek and R. Koekoek: On a differential equation for Koornwinder's generalized Laguerre polynomials. Proc. Amer. Math. Soc. 112, 1991, 1045-1054.MR 91j:33008
10.: J. Koekoek , R. Koekoek and H. Bavinck: On differential equations for Sobolev-type Laguerre polynomials. TUDelft Fac. Techn. Math. & Inf. Report 95-79. To appear in Trans. Amer. Math. Soc.
11.: R. Koekoek: Generalizations of the classical Laguerre polynomials. J. Math. Anal. Appl. 153 (1990), 576-590. MR 92d:33016
12.: R. Koekoek: Generalizations of the classical Laguerre polynomials and some -analogues. Thesis Delft University of Technology 1990.
13.: T.H. Koornwinder: Orthogonal polynomials with weight function $\left( 1-x\right) ^\alpha \left( 1+x\right) ^\beta +M\delta (x+1)+N\delta (x-1).$ Canad. Math. Bull. 27(2), 1984, 205-214.MR 85i:33011
14.: F. Marcellán and A. Ronveaux: On a class of polynomials orthogonal with respect to a Sobolev inner product. Indag. Math. N.S. 1 (1990), 451-464.MR 92f:42029

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33C45, 34A35

H. Bavinck
Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, Mekelweg 4, 2628 CD Delft, The Netherlands
Email: bavinck@twi.tudelft.nl

DOI: 10.1090/S0002-9939-97-04091-4
PII: S 0002-9939(97)04091-4
Keywords: Differential operators, Sobolev type Laguerre polynomials
Received by editor(s): June 27, 1996
Communicated by: Christopher D. Sogge
Copyright of article: Copyright 1997, American Mathematical Society

H. Bavinck, Differential and difference operators having orthogonal polynomials with linear perturbations as eigenfunctions,J. Comp. Appl. Math. 92(1998), 85-95. (English)

H. Bavinck, Differential and difference operators having orthogonal polynomials with two linear perturbations as eigenfunctions,J. Comp. Appl. Math. 92(1998), 85-95. (English)

H. Bavinck, On the sum of the coefficients of certain linear differential operators,J. Comp. Appl. Math. 89(1998), 213-217. (English)

K. Srinivasa Rao, R. Jagannathan, G. Vanden Berghe, J. Van der Jeugt,Differential operators having Laguerre type and Sobolev type Laguerre polynomials as eigenfunctions: a survey,Special Functions and Differential Equations (Madras, India,January 13-24, 1997), Allied PublishersPrivate Lt. , New Delhi,India, 1998, pp. 102-118. (English)

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS