Letter to the Editor
On a linear perturbation of the Laguerre operator
H. Bavinck∗
Delft University of Technology, Faculty of Information Technology and Systems, Mekelweg 4, 2628 CD Delft, The Netherlands
Received 8 January 1999
Abstract
In Koekoek and Koekoek (On a dierential equation for Koornwinder’s generalized Laguerre polynomials, Proc. Amer. Math. Soc. 112 (1991) 1045 –1054) a dierential equation is introduced, which is of nite order if is a nonnegative integer and is of innite order for other values of ¿−1; and can be interpreted as a linear perturbation of the dierential operator for Laguerre polynomials, having orthogonal polynomials, generalizations of Laguerre polynomials (Laguerre type polynomials), as eigenfunctions. In this letter we give two new representations of this operator (with the perturbation operator in factorized form) in the nite order case and we mention some properties of this operator.
c
1999 Elsevier Science B.V. All rights reserved.
MSC:33C45; 34A35
Keywords:Orthogonal polynomial; Laguerre polynomial; Dierential operator
1. Introduction
In [11] Koornwinder introduced the polynomials {L; M n (x)}
∞
n=0; dened by
L; M
n (x) =L(n)(x) +MQ(n)(x) (1.1)
with Q(0)(x) = 0 and for n∈ {1;2;3; : : :}
Q(n)(x) =
n+
n−1
L(n)(x) +
n+
n
DL(n)(x)
=−
n+
n
x
+ 1L
(+2)
n−1 (x); (1.2)
∗Tel.: +31 15 278 5822; fax: +31 15 278 7245.
E-mail address: [email protected] (H. Bavinck)
(D=d
dx); which are orthogonal with respect to the inner product (f; g) = 1
They are usually called Laguerre type polynomials, which are studied in great detail in [9]. For ∈ {0;1;2} they have been considered earlier (see [12–14], where dierential equations were found for these polynomials). In [7] (see also [10]) Koekoek and Koekoek discovered a dierential operator having the Laguerre type polynomials {L; M
n (x)}
∞
n=0 ( ¿−1) as eigenfunctions, generalizing the
earlier results.
Since for the classical Laguerre polynomials {L() n (x)}
an operator of the form
L()+MA()
By substituting Eq. (1.1) into Eq. (1.7) and by equating the coecients of M and M2 on both
sides, it is derived that
A()L(n)(x) =n()L(n)(x)− systems of equations for the unknown constants {()
n }
∞
n=1 and the unknown functions {ai(x;)}
∞
i=1;
have a unique solution given by
(n)=
2. Factorizations of A()
We study the case that the operator A() is of nite order. Therefore in the rest of this letter we
will assume that is a nonnegative integer.
2.1. The rst factorization
where I denotes the identity operator. The polynomials {Q() n (x)}
∞
n=1 are a basis for the space of all
polynomials divisible by x: In order to conclude that
A()= 1
it is sucient to prove the following
Lemma 2.1. The operator Q+1
which also can be seen from Eq. (1.12).
2.2. The second factorization
Another factorization of the operator A() can be obtained from Eq. (1.9). If we remember
all the Laguerre polynomials (for n= 0 this is trivial) as the operator A() and therefore coincides
some calculations we nd a second factorization of A():
A()= x
• The operator A() maps polynomials of degreen(n¿1) onto the subspace of polynomials of degree
n that have no constant term. By Eq. (1.10) the polynomials {Q() n (x)}
∞
n=1 are eigenfunctions of
this operator.
This was proved in [8]. Another proof, which also holds in a more general situation, is given in [3]. There it appeared that the real reason for this property is that ex is an eigenfunction of L():
Yet another proof can be given by using formula (2.2). In fact
and
(L; Mn (x);(L()+M
A())L; Mm (x)) = (m+Mm())(L; Mn (x); L; Mm (x)) = 0 by the orthogonality. With respect to the Laguerre inner product
hf; gi= 1 (+ 1)
Z ∞
0
f(x)g(x)xe−x
dx
this is still true for all polynomials which have no constant term, which is rather obvious, since there the two inner products coincide, but
h(L()+MA())L; Mn (x);1i= (n+M(n))hL; Mn (x);1i
= (n+M(n))hLn()(x) +MQn()(x);1i
=−
n+
n
(n+M() n )M
+ 1 hxL
(+2) n−1 (x);1i
=−(n+)!
n! (n+M
() n )M and
hL; Mn (x);(L()+MA())1i= 0:
This shows that the statement in [6], that the dierential operator L()+MA() is symmetrizable
with symmetry factor xe−x is not completely correct.
• Generalizations of the operatorA() to cases, where the inner product contains derivatives (Sobolev
type Laguerre polynomials), are derived in [2,4,8].
Acknowledgements
The author wishes to thank J. Koekoek for valuable remarks.
References
[1] H. Bavinck, A direct approach to Koekoek’s dierential equation for generalized Laguerre polynomials, Acta Math. Hungar. 66 (3) (1995) 247–253.
[2] H. Bavinck, Dierential operators having Sobolev type Laguerre polynomials as eigenfunctions, Proc. Amer. Math. Soc. 125 (1997) 3561–3567.
[3] H. Bavinck, On the sum of the coecients of certain dierential operators, J. Comput. Appl. Math. 89 (1998) 213–217.
[4] H. Bavinck, Dierential operators having Sobolev type Laguerre polynomials as eigenfunctions: new developments, in press.
[5] W.N. Everitt, L.L. Littlejohn, Orthogonal polynomials and spectral theory: a survey, in: C. Brezinski, L. Gori, A. Ronveaux (Eds.), Orthogonal Polynomials and their Applications, IMACS Annals on Computing and Applied Mathematics, vol. 9, J.C. Baltzer AG, Basel, 1991, pp. 21–55.
[7] J. Koekoek, R. Koekoek, On a dierential equation for Koornwinder’s generalized Laguerre polynomials, Proc. Amer. Math. Soc. 112 (1991) 1045–1054.
[8] J. Koekoek, R. Koekoek, H. Bavinck, On dierential equations for Sobolev-type Laguerre polynomials, TUDelft Fac. Techn. Math. & Inf. Report 95-79, Trans. Amer. Math. Soc. 350 (1998) 347–393.
[9] R. Koekoek, Koornwinder’s Laguerre polynomials, Delft Progress Rep. 12 (1988) 393–403.
[10] R. Koekoek, Generalizations of the classical Laguerre polynomials and some q-analogues, Thesis, Delft University of Technology, 1990.
[11] T.H. Koornwinder, Orthogonal polynomials with weight function (1−x)(1 +x)+M(x+ 1) +N(x−1), Canad.
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[12] A.M. Krall, L.L. Littlejohn, On the classication of dierential equations having orthogonal polynomial solutions II, Ann. Mat. Pura Appl. (4) 149 (1987) 77–102.
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