Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol106.Issue1.1999:
Journal of Computational and Applied Mathematics 106 (1999) 197–202
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 di erential equation for Koornwinder’s generalized Laguerre polynomials, Proc. Amer.
Math. Soc. 112 (1991) 1045 –1054) a di erential equation is introduced, which is of nite order if is a nonnegative
integer and is of in nite order for other values of ¿ − 1; and can be interpreted as a linear perturbation of the
di erential 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; Di erential operator
1. Introduction
In [11] Koornwinder introduced the polynomials {L ;n M (x)}∞
n=0 ; de ned by
L ;n M (x) = Ln( ) (x) + MQn( ) (x)
(1.1)
with Q0( ) (x) = 0 and for n ∈ {1; 2; 3; : : :}
n+
n+
Ln( ) (x) +
DLn( ) (x)
n−1
n
n+
x
L( +2) (x);
=−
n
+ 1 n−1
Qn( ) (x) =
∗
Tel.: +31 15 278 5822; fax: +31 15 278 7245.
E-mail address: [email protected] (H. Bavinck)
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 7 8 - 3
(1.2)
198
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
(D= ddx ); which are orthogonal with respect to the inner product
Z ∞
1
(f; g) =
f(x)g(x)x e−x d x + Mf(0)g(0); M ¿0; ¿ − 1:
(1.3)
( + 1) 0
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 di erential equations were found
for these polynomials). In [7] (see also [10]) Koekoek and Koekoek discovered a di erential operator
having the Laguerre type polynomials {L ;n M (x)}∞
n=0 ( ¿ − 1) as eigenfunctions, generalizing the
earlier results.
Since for the classical Laguerre polynomials {Ln( ) (x)}∞
n=0 it is known that
L( ) Ln( ) (x) = nLn( ) (x)
(1.4)
L( ) := − xD2 − ( + 1 − x)D;
(1.5)
with
an operator of the form
L( ) + M A( )
is considered, where
A( ) :=
∞
X
ai (x; )Di ;
(1.6)
i=1
and numbers { n( ) }∞
n=0 such that
[L( ) + M A( ) ]L ;n M (x) = [n + M n( ) ]L ;n M (x)
for n ∈ {0; 1; 2; : : :}: It is easy to derive that
L
( )
DLn( ) (x)
−
nDLn( ) (x)
=
D2 Ln( ) (x)
−
0( )
(1.7)
= 0 and that
DLn( ) (x)
( +2)
= Ln−1
(x):
(1.8)
By substituting Eq. (1.1) into Eq. (1.7) and by equating the coecients of M and M 2 on both
sides, it is derived that
n+
( +2)
Ln−1
(x);
(1.9)
A( ) Ln( ) (x) = n( ) Ln( ) (x) −
n
A( ) Qn( ) (x) = n( ) Qn( ) (x);
(1.10)
for all real x and n ∈ {1; 2; 3; : : :}: Koekoek and Koekoek succeeded in showing that these two
∞
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+ +1
n( ) =
; n ∈ {1; 2; 3; : : :};
(1.11)
n−1
ai (x ; ) =
i
1X
(−1)i+j
i! j=1
+1
j−1
+2
( + 3)i−j xj ;
i−j
i ∈ {1; 2; 3; : : :}:
(1.12)
From Eq. (1.12) it follows at once that the di erential operator A( ) is of order 2 + 4 if is a
nonnegative integer and of in nite order if is not a nonnegative integer. A more direct approach to
this result is given in [1] and the connection to spectral theory of di erential operators is discussed
in [5].
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
199
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
We can write
n( ) =
+1
Y
1
(n + j );
( + 2)! j=0
and since by Eqs. (1.2), (1.4) and (1.8) we have
+1
L( ) +
I Qn( ) (x) = nQn( ) (x);
x
it follows from Eq. (1.10) that for all n ∈ N
A( ) Qn( ) (x) =
1
( + 2)!
+1
Y
L( ) +
j=0
+1
+ j I Qn( ) (x);
x
where I denotes the identity operator. The polynomials {Qn( ) (x)}∞
n=1 are a basis for the space of all
polynomials divisible by x: In order to conclude that
A
( )
+1
Y
+1
1
+j I ;
L( ) +
=
( + 2)! j=0
x
(2.1)
it is sucient to prove the following
Lemma 2.1. The operator
Q +1
j=0 (L
( )
+ ( +1
+ j )I ) annihilates constants.
x
Proof. It is easy to verify that for all j ∈ {0; 1; 2; : : :}
+1
1
( j + 1)(−j + + 1)
L +
+j I j =
:
x
x
xj+1
By applying this consecutively for j = 0; 1; : : : ; a + 1 we obtain the desired result.
( )
From Eq. (2.1) it follows immediately that A( ) is of order 2 + 4 and that
(−x) +2
;
( + 2)!
which also can be seen from Eq. (1.12).
a2 +4 (x; ) =
2.2. The second factorization
Another factorization of the operator A( ) can be obtained from Eq. (1.9). If we remember
Eq. (1.8) it is easy to construct a linear di erential operator which has the same working on
200
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
all the Laguerre polynomials (for n = 0 this is trivial) as the operator A( ) and therefore coincides
with A( ) : In fact
A( ) =
=
+1
Y
Y
1
1
(L( ) + jI ) − (D2 − D) (L( ) + jI )
( + 2)! j=0
!
j=1
Y
1
[(L( ) + ( + 1)I )L( ) − ( + 2)( + 1)(D2 − D)] (L( ) + jI ):
( + 2)!
j=1
The fourth order operator [L( ) + ( + 1)I ]L( ) − ( + 2)( + 1)(D2 − D) can be factorized. After
some calculations we nd a second factorization of A( ) :
Y
x
( )
2
2
[xD + (2 + 4 − x)D − ( + 2)I ](D − D) (L( ) + jI ):
(2.2)
A =
( + 2)!
j=1
In the case that = 0; we interpret
Q
j=1 (L
( )
+ jI ) = I :
3. Properties of the operator A( )
• The operator A( ) maps polynomials of degree n(n¿1) onto the subspace of polynomials of degree
n that have no constant term. By Eq. (1.10) the polynomials {Qn( ) (x)}∞
n=1 are eigenfunctions of
this operator.
2 +4
• The coecients {ai (x; )}i=1
have the interesting property that
2 +4
X
ai (x; ) = 0:
(3.1)
i=1
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 e x is an eigenfunction of L( ) :
Yet another proof can be given by using formula (2.2). In fact
Y
(L( ) + jI )ex =
Y
( + 1 + j )ex ;
j=1
j=1
showing that
2
(D − D)
Y
(L( ) + jI )ex = 0:
j=1
It follows that A( ) ex = 0; which implies Eq. (3.1).
• The operator L( ) + M A( ) is self-adjoint with respect to the inner product (1.3) for all f; g∈C 2 +4 ;
∞
such that (1.3) exists. Since the polynomials are dense in C 2 +4 and the polynomials {L ;M
n (x )}n=0
are a basis for the polynomials, it suces to show that for all n; m ∈ {0; 1; 2; : : :}
((L( ) + M A( ) )L ;n M (x); L ;m M (x)) = (L ;n M (x); (L( ) + M A( ) )L ;m M (x)):
For n = m this is obvious and for n 6= m
((L( ) + M A( ) )L ;n M (x); L ;m M (x)) = (n + M n( ) )(L ;n M (x); L ;m M (x)) = 0
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
201
and
(L ;n M (x); (L( ) + M A( ) )L ;m M (x)) = (m + M m( ) )(L ;n M (x); L ;m M (x)) = 0
by the orthogonality. With respect to the Laguerre inner product
hf; gi =
1
( + 1)
Z
∞
f(x)g(x)x e−x d x
0
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( ) + M A( ) )L ;n M (x); 1i = (n + M n( ) )hL ;n M (x); 1i
= (n + M n( ) )hLn( ) (x) + MQn( ) (x); 1i
n+
n
(n + M n( ) )M
( +2)
hxLn−1
(x); 1i
+1
=−
=−
(n + )!
(n + M n( ) )M
n!
and
hL ;n M (x); (L( ) + M A( ) )1i = 0:
This shows that the statement in [6], that the di erential operator L( ) + M A( ) is symmetrizable
with symmetry factor x e−x is not completely correct.
• Generalizations of the operator A( ) 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 di erential equation for generalized Laguerre polynomials, Acta Math.
Hungar. 66 (3) (1995) 247–253.
[2] H. Bavinck, Di erential 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 di erential operators, J. Comput. Appl. Math. 89 (1998)
213–217.
[4] H. Bavinck, Di erential 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.
[6] W.N. Everitt, L.L. Littlejohn, R. Wellman, The symmetric form of the Koekoeks’ Laguerre type di erential equation,
J. Comput. Appl. Math. 57 (1995) 115–121.
202
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
[7] J. Koekoek, R. Koekoek, On a di erential equation for Koornwinder’s generalized Laguerre polynomials, Proc. Amer.
Math. Soc. 112 (1991) 1045–1054.
[8] J. Koekoek, R. Koekoek, H. Bavinck, On di erential 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.
Math. Bull. 27 (2) (1984) 205–214.
[12] A.M. Krall, L.L. Littlejohn, On the classi cation of di erential equations having orthogonal polynomial solutions II,
Ann. Mat. Pura Appl. (4) 149 (1987) 77–102.
[13] H.L. Krall, On orthogonal polynomials satisfying a certain fourth order di erential equation, The Pennsylvania State
College Studies, No. 6, The Pennsylvania State College, State College, PA, 1940.
[14] L.L. Littlejohn, An application of a new theorem on orthogonal polynomials and di erential equations, Quaestiones
Math. 10 (1986) 49–61.
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 di erential equation for Koornwinder’s generalized Laguerre polynomials, Proc. Amer.
Math. Soc. 112 (1991) 1045 –1054) a di erential equation is introduced, which is of nite order if is a nonnegative
integer and is of in nite order for other values of ¿ − 1; and can be interpreted as a linear perturbation of the
di erential 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; Di erential operator
1. Introduction
In [11] Koornwinder introduced the polynomials {L ;n M (x)}∞
n=0 ; de ned by
L ;n M (x) = Ln( ) (x) + MQn( ) (x)
(1.1)
with Q0( ) (x) = 0 and for n ∈ {1; 2; 3; : : :}
n+
n+
Ln( ) (x) +
DLn( ) (x)
n−1
n
n+
x
L( +2) (x);
=−
n
+ 1 n−1
Qn( ) (x) =
∗
Tel.: +31 15 278 5822; fax: +31 15 278 7245.
E-mail address: [email protected] (H. Bavinck)
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 7 8 - 3
(1.2)
198
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
(D= ddx ); which are orthogonal with respect to the inner product
Z ∞
1
(f; g) =
f(x)g(x)x e−x d x + Mf(0)g(0); M ¿0; ¿ − 1:
(1.3)
( + 1) 0
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 di erential equations were found
for these polynomials). In [7] (see also [10]) Koekoek and Koekoek discovered a di erential operator
having the Laguerre type polynomials {L ;n M (x)}∞
n=0 ( ¿ − 1) as eigenfunctions, generalizing the
earlier results.
Since for the classical Laguerre polynomials {Ln( ) (x)}∞
n=0 it is known that
L( ) Ln( ) (x) = nLn( ) (x)
(1.4)
L( ) := − xD2 − ( + 1 − x)D;
(1.5)
with
an operator of the form
L( ) + M A( )
is considered, where
A( ) :=
∞
X
ai (x; )Di ;
(1.6)
i=1
and numbers { n( ) }∞
n=0 such that
[L( ) + M A( ) ]L ;n M (x) = [n + M n( ) ]L ;n M (x)
for n ∈ {0; 1; 2; : : :}: It is easy to derive that
L
( )
DLn( ) (x)
−
nDLn( ) (x)
=
D2 Ln( ) (x)
−
0( )
(1.7)
= 0 and that
DLn( ) (x)
( +2)
= Ln−1
(x):
(1.8)
By substituting Eq. (1.1) into Eq. (1.7) and by equating the coecients of M and M 2 on both
sides, it is derived that
n+
( +2)
Ln−1
(x);
(1.9)
A( ) Ln( ) (x) = n( ) Ln( ) (x) −
n
A( ) Qn( ) (x) = n( ) Qn( ) (x);
(1.10)
for all real x and n ∈ {1; 2; 3; : : :}: Koekoek and Koekoek succeeded in showing that these two
∞
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+ +1
n( ) =
; n ∈ {1; 2; 3; : : :};
(1.11)
n−1
ai (x ; ) =
i
1X
(−1)i+j
i! j=1
+1
j−1
+2
( + 3)i−j xj ;
i−j
i ∈ {1; 2; 3; : : :}:
(1.12)
From Eq. (1.12) it follows at once that the di erential operator A( ) is of order 2 + 4 if is a
nonnegative integer and of in nite order if is not a nonnegative integer. A more direct approach to
this result is given in [1] and the connection to spectral theory of di erential operators is discussed
in [5].
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
199
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
We can write
n( ) =
+1
Y
1
(n + j );
( + 2)! j=0
and since by Eqs. (1.2), (1.4) and (1.8) we have
+1
L( ) +
I Qn( ) (x) = nQn( ) (x);
x
it follows from Eq. (1.10) that for all n ∈ N
A( ) Qn( ) (x) =
1
( + 2)!
+1
Y
L( ) +
j=0
+1
+ j I Qn( ) (x);
x
where I denotes the identity operator. The polynomials {Qn( ) (x)}∞
n=1 are a basis for the space of all
polynomials divisible by x: In order to conclude that
A
( )
+1
Y
+1
1
+j I ;
L( ) +
=
( + 2)! j=0
x
(2.1)
it is sucient to prove the following
Lemma 2.1. The operator
Q +1
j=0 (L
( )
+ ( +1
+ j )I ) annihilates constants.
x
Proof. It is easy to verify that for all j ∈ {0; 1; 2; : : :}
+1
1
( j + 1)(−j + + 1)
L +
+j I j =
:
x
x
xj+1
By applying this consecutively for j = 0; 1; : : : ; a + 1 we obtain the desired result.
( )
From Eq. (2.1) it follows immediately that A( ) is of order 2 + 4 and that
(−x) +2
;
( + 2)!
which also can be seen from Eq. (1.12).
a2 +4 (x; ) =
2.2. The second factorization
Another factorization of the operator A( ) can be obtained from Eq. (1.9). If we remember
Eq. (1.8) it is easy to construct a linear di erential operator which has the same working on
200
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
all the Laguerre polynomials (for n = 0 this is trivial) as the operator A( ) and therefore coincides
with A( ) : In fact
A( ) =
=
+1
Y
Y
1
1
(L( ) + jI ) − (D2 − D) (L( ) + jI )
( + 2)! j=0
!
j=1
Y
1
[(L( ) + ( + 1)I )L( ) − ( + 2)( + 1)(D2 − D)] (L( ) + jI ):
( + 2)!
j=1
The fourth order operator [L( ) + ( + 1)I ]L( ) − ( + 2)( + 1)(D2 − D) can be factorized. After
some calculations we nd a second factorization of A( ) :
Y
x
( )
2
2
[xD + (2 + 4 − x)D − ( + 2)I ](D − D) (L( ) + jI ):
(2.2)
A =
( + 2)!
j=1
In the case that = 0; we interpret
Q
j=1 (L
( )
+ jI ) = I :
3. Properties of the operator A( )
• The operator A( ) maps polynomials of degree n(n¿1) onto the subspace of polynomials of degree
n that have no constant term. By Eq. (1.10) the polynomials {Qn( ) (x)}∞
n=1 are eigenfunctions of
this operator.
2 +4
• The coecients {ai (x; )}i=1
have the interesting property that
2 +4
X
ai (x; ) = 0:
(3.1)
i=1
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 e x is an eigenfunction of L( ) :
Yet another proof can be given by using formula (2.2). In fact
Y
(L( ) + jI )ex =
Y
( + 1 + j )ex ;
j=1
j=1
showing that
2
(D − D)
Y
(L( ) + jI )ex = 0:
j=1
It follows that A( ) ex = 0; which implies Eq. (3.1).
• The operator L( ) + M A( ) is self-adjoint with respect to the inner product (1.3) for all f; g∈C 2 +4 ;
∞
such that (1.3) exists. Since the polynomials are dense in C 2 +4 and the polynomials {L ;M
n (x )}n=0
are a basis for the polynomials, it suces to show that for all n; m ∈ {0; 1; 2; : : :}
((L( ) + M A( ) )L ;n M (x); L ;m M (x)) = (L ;n M (x); (L( ) + M A( ) )L ;m M (x)):
For n = m this is obvious and for n 6= m
((L( ) + M A( ) )L ;n M (x); L ;m M (x)) = (n + M n( ) )(L ;n M (x); L ;m M (x)) = 0
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
201
and
(L ;n M (x); (L( ) + M A( ) )L ;m M (x)) = (m + M m( ) )(L ;n M (x); L ;m M (x)) = 0
by the orthogonality. With respect to the Laguerre inner product
hf; gi =
1
( + 1)
Z
∞
f(x)g(x)x e−x d x
0
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( ) + M A( ) )L ;n M (x); 1i = (n + M n( ) )hL ;n M (x); 1i
= (n + M n( ) )hLn( ) (x) + MQn( ) (x); 1i
n+
n
(n + M n( ) )M
( +2)
hxLn−1
(x); 1i
+1
=−
=−
(n + )!
(n + M n( ) )M
n!
and
hL ;n M (x); (L( ) + M A( ) )1i = 0:
This shows that the statement in [6], that the di erential operator L( ) + M A( ) is symmetrizable
with symmetry factor x e−x is not completely correct.
• Generalizations of the operator A( ) 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 di erential equation for generalized Laguerre polynomials, Acta Math.
Hungar. 66 (3) (1995) 247–253.
[2] H. Bavinck, Di erential 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 di erential operators, J. Comput. Appl. Math. 89 (1998)
213–217.
[4] H. Bavinck, Di erential 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.
[6] W.N. Everitt, L.L. Littlejohn, R. Wellman, The symmetric form of the Koekoeks’ Laguerre type di erential equation,
J. Comput. Appl. Math. 57 (1995) 115–121.
202
H. Bavinck / Journal of Computational and Applied Mathematics 106 (1999) 197–202
[7] J. Koekoek, R. Koekoek, On a di erential equation for Koornwinder’s generalized Laguerre polynomials, Proc. Amer.
Math. Soc. 112 (1991) 1045–1054.
[8] J. Koekoek, R. Koekoek, H. Bavinck, On di erential 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.
Math. Bull. 27 (2) (1984) 205–214.
[12] A.M. Krall, L.L. Littlejohn, On the classi cation of di erential equations having orthogonal polynomial solutions II,
Ann. Mat. Pura Appl. (4) 149 (1987) 77–102.
[13] H.L. Krall, On orthogonal polynomials satisfying a certain fourth order di erential equation, The Pennsylvania State
College Studies, No. 6, The Pennsylvania State College, State College, PA, 1940.
[14] L.L. Littlejohn, An application of a new theorem on orthogonal polynomials and di erential equations, Quaestiones
Math. 10 (1986) 49–61.