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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**H. Bavinck :*A direct approach to Koekoek's differential equation for generalized Laguerre polynomials.*Acta Mathematica Hungarica, 1995, 247-253. MR**66****96a:33009****2.**H. Bavinck :*A difference operator of infinite order with Sobolev-type Charlier polynomials as eigenfunctions.*Indagationes Mathematicae, (N.S.)**7**(3), 1996, 281-291.**3.**H. Bavinck & H. van Haeringen :*Difference equations for generalized Meixner polynomials.*Journal of Mathematical Analysis and Applications**184**, 1994, 453-463. MR**95h:33004****4.**H. Bavinck & R. Koekoek :*On a difference equation for generalizations of Charlier polynomials.*Journal of Approximation Theory**81**, 1995, 195-206. MR**96h:33002****5.**W.N. Everitt, L.L. Littlejohn & R. Wellman :*The symmetric form of the Koekoeks' Laguerre type differential equation.*Journal of Computational and Applied Mathematics**57**, 1995, 115-121. MR**96j:34148****6.**I.H. Jung, K.H. Kwon, D.W. Lee & L.L. Littlejohn :*Sobolev orthogonal polynomials and spectral differential equations.*Transactions of the American Mathematical Society**347**, 1995, 3629-3643. CMP**95:14****7.**I.H. Jung, K.H. Kwon, D.W. Lee & L.L. Littlejohn :*Differential equations and Sobolev orthogonality.*Journal of Computational and Applied Mathematics**65**, 1995, 173-180. MR**97a:33019****8.**J. Koekoek & R. Koekoek :*On a differential equation for Koornwinder's generalized Laguerre polynomials.*Proceedings of the American Mathematical Society**112**, 1991, 1045-1054. MR**91j:33008****9.**R. Koekoek :*Generalizations of the classical Laguerre polynomials and some q-analogues.*Delft University of Technology, Thesis, 1990.**10.**R. Koekoek :*The search for differential equations for orthogonal polynomials by using computers.*Delft University of Technology, report no.**91-55**, 1991.**11.**R. Koekoek :*The search for differential equations for certain sets of orthogonal polynomials.*Journal of Computational and Applied Mathematics**49**, 1993, 111-119. MR**95m:33008****12.**R. Koekoek & H.G. Meijer :*A generalization of Laguerre polynomials.*SIAM Journal on Mathematical Analysis**24**, 1993, 768-782. MR**94b:33007****13.**T.H. Koornwinder :*Orthogonal polynomials with weight function .*Canadian Mathematical Bulletin**27**(2), 1984, 205-214. MR**85i:33011****14.**A.M. Krall :*Orthogonal polynomials satisfying fourth order differential equations.*Proceedings of the Royal Society of Edinburgh Sect. A**87**, 1981, 271-288. MR**82d:33021****15.**H.L. Krall :*Certain differential equations for Tchebycheff polynomials.*Duke Mathematical Journal**4**, 1938, 705-718.**16.**H.L. Krall :*On orthogonal polynomials satisfying a certain fourth order differential equation.*The Pennsylvania State College Studies, No. 6, 1940. MR**2:98a****17.**H.L. Krall & I.M. Sheffer :*Differential equations of infinite order for orthogonal polynomials.*Annali di Matematica Pura ed Applicata (4)**74**, 1966, 135-172. MR**34:6260****18.**Y.L. Luke :*The Special Functions and Their Approximations II.*Academic Press, New York, 1969. MR**40:2909****19.**C.S. Meijer :*Expansion theorems for the -function I.*Indagationes Mathematicae**14**, 1952, 369-379. MR**14:469e**

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

Retrieve articles in all journals with MSC (1991): 33C45, 34A35

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