Sobolev orthogonality for the Gegenbauer
polynomials
{
C
(−N+1=2)n
}
n¿0Mara Alvarez de Moralesa;1, Teresa E. Perezb;2, Miguel A. Pi ˜nar2;b;∗
aDepartamento de Matematica Aplicada, Universidad de Granada, 18071 Granada, Spain bDepartamento de Matematica Aplicada and Instituto Carlos I de Fsica Teorica y Computacional,
Universidad de Granada, 18071 Granada, Spain
Received 13 November 1997; received in revised form 15 July 1998
Abstract
In this work, we obtain the property of Sobolev orthogonality for the Gegenbauer polynomials {C(−N+1=2)
n }n¿0, with N¿1 a given nonnegative integer, that is, we show that they are orthogonal with respect to some inner product involving derivatives. The Sobolev orthogonality can be used to recover properties of these Gegenbauer polynomials. For instance, we can obtain linear relations with the classical Gegenbauer polynomials. c1998 Elsevier Science B.V. All rights reserved.
Keywords:Gegenbauer polynomials; Nondiagonal Sobolev inner product; Sobolev orthogonal polynomials
1. Introduction
For any value of the parameter ¿−1
2 monic Gegenbauer polynomials {C ()
n }n¿0 can be dened by their explicit representation (see [7, p. 84])
C()
n (x) = n! 2n (n+)
[n=2]
X
m=0
(−1)m (n−m+) m!(n−2m)! (2x)
n−2m; ¿−1
2:
Simplifying the values of the Gamma function in this expression we get
C()
n (x) = n! 2n
[n=2]
X
m=0
(−1)m
(n+−m)mm!(n−2m)!
(2x)n−2m; (1)
∗Corresponding author. E-mail: [email protected].
1Partially supported by Junta de Andaluca, Grupo de Investigacion FQM 0229.
2Partially supported by Junta de Andaluca, Grupo de Investigacion FQM 0229, DGES under grant PB 95-1205 and INTAS-93-0219-ext.
where (n+−m)m denotes the usual Pochhammer symbol dened by
(b)0= 1; (b)n=b(b+ 1)· · ·(b+n−1); b∈R; ∀n¿1:
In this way, we must notice that expression (1) is valid for every real value of the parameter 6=−l, with l= 0;1;2; : : : and, therefore, it can be used to dene Gegenbauer polynomials for 6= 0;−1;−2; : : : .
From the explicit representation (1) we can deduce that the monic Gegenbauer polynomials {C()
n }n¿0 is orthogonal with respect to a quasi-denite linear functional. Besides, if −1
2¡ the functional is positive denite and the polynomials are orthogonal with respect to the weight (1−x2)−1=2 on the interval [−1;1]. For =−1
value of n, no orthogonality results can be deduced from Favard’s Theorem.
The main objective of this paper is to obtain orthogonality properties for the Gegenbauer poly-nomials {C(−N+1=2)
n }n¿0. In fact, we are going to show that they are orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product.
Similar results for dierent families of classical polynomials have been the subject of an increasing number of papers. For instance, Kwon and Littlejohn, in [3], established the orthogonality of the generalized Laguerre polynomials {L(−k)
n }n¿0; k¿1, with respect to a Sobolev inner product of the
with A a symmetric k ×k real matrix. In [4], the same authors showed that Jacobi polynomials {P(−1;−1)
n }n¿0, are orthogonal with respect to the inner product
Later, in [5], Perez and Pi˜nar gave a unied approach to the orthogonality of the generalized Laguerre polynomials {L()
n }n¿0, for any real value of the parameter by proving their orthogo-nality with respect to a Sobolev nondiagonal inner product. In [6], they showed how to use this orthogonality to obtain dierent properties of the generalized Laguerre polynomials.
Alfaro et al., in [1], studied sequences of polynomials which are orthogonal with respect to a Sobolev bilinear form dened by
B(N)
where u is a quasi-denite linear functional on the linear space P of real polynomials, c is a real number, N is a positive integer number, and A is a symmetric N ×N real matrix such that each of its principal submatrices is regular. In particular, they deduced that the Jacobi polynomials {P(−N; )
n }n¿0, for +N not a negative integer are orthogonal with respect to (3). In this case, I= [−1;1], u is the Jacobi functional dened from the weight function (0; +N)(x) = (1 +x)+N, c= 1 and A=Q−1D(Q−1)T, where D is a regular diagonal matrix and Q is the matrix of the derivatives of the Jacobi polynomials at the point 1.
Now, we are going to describe the structure of the paper. In Section 2, from the explicit repre-sentation of the monic Gegenbauer polynomials, {C()
n }n¿0, for ∈R; 6= 0;−1;−2; : : : we deduce some of their usual properties, namely the three-term recurrence relation, the dierentiation property, the second order dierential equation, : : : .
Next, in Section 3, we show that the Gegenbauer polynomials {C(−N+1=2)
n }n¿0, are orthogonal with
respect to the Sobolev inner product dened by
(f; g)S = f(1); f
where f and g are two arbitrary polynomials and N¿1 is a positive integer number.
The last section of the paper is devoted to the study of a linear dierential operator, F, which
is dened on the linear space of the real polynomials P, and is symmetric with respect to the Sobolev inner product (4). By using this operator, we can establish several linear relations between Gegenbauer polynomials {C(−N+1=2)
2. The Gegenbauer polynomials
For a given real number ¿−1
2, the explicit representation of the nth monic Gegenbauer polyno-mial is given by
Cn()(x) = n! 2n (n+)
[n=2]
X
m=0
(−1)m (n−m+) m!(n−2m)! (2x)
n−2m; n¿0; (5)
(see [7, p. 84]). As it is well known, for ¿−1
2 the sequence {C ()
n }n¿0 constitutes a monic orthog-onal polynomial sequence associated with the weight function (x) = (1−x2)−1=2 on the interval [−1;1].
After simplication, expression (5) reads
C()
n (x) = n! 2n
[n=2]
X
m= 0
(−1)m
(+n−m)mm!(n−2m)!
(2x)n−2m; n¿0: (6)
We can observe that for every value of the parameter 6=−l; l= 0;1;2; : : : ; expression (6) denes a monic polynomial of exact degree n. In this way, for 6=−l; l= 0;1;2; : : :we can dene a family of monic polynomials {C()
n }n¿0 which is a basis of the linear space of the real polynomials. From now on, these polynomials will be called generalized Gegenbauer polynomials.
Very simple manipulations of the explicit representation show that the main part of the alge-braic properties of the classical Gegenbauer polynomials holds for every value of the parameter 6=−l; l= 0;1;2; : : : ; as we show in the following proposition.
Proposition 1. Let be an arbitrary real number with 6=−l; forl= 0;1;2; : : :. Then;the Gegen-bauer polynomials {C()
n }n¿0; satisfy the following properties: (i) Symmetry
Cn()(−x) = (−1)nCn()(x): (7)
(ii) Three-term recurrence relation
C−(1)(x) = 0; C ()
0 (x) = 1;
xCn()(x) =Cn(+1)(x) +n()Cn(−)1(x); n¿0; (8)
where
(n)= n(n+ 2−1) 4(n+)(n+−1):
(iii) Dierentiation property
DC()
n (x) =nC
(+1)
n−1 (x); (9)
(iv) Second-order dierential equation
(1−x2)y′′
−(2+ 1)xy′
+n(n+ 2)y= 0;
where y=C()
n (x):
We can observe that the parameter ()
n in the three-term recurrence relation does not vanish for
every value of 6=−N + 1
2, for N= 1;2;3; : : : . Thus, from Favard’s Theorem (see [2, p. 21]) we deduce that the generalized Gegenbauer polynomials {C()
n }n¿0, 6=−N +12 for N= 1;2;3; : : : ;
are orthogonal with respect to a certain moment functional. For ¿−1
2 this moment functional is positive denite. However, since (2−NN+1=2)= 0, no orthogonality properties can be deduced from Favard’s Theorem for the sequence of polynomials {C(−N+1=2)
n }n¿0 for N= 1;2;3; : : : . In order to obtain orthogonality properties for the Gegenbauer polynomials {C(−N+1=2)
n }n¿0, for
N= 1;2;3; : : : ; we have to emphasize some of their interesting properties.
Proposition 2. Let N be a given positive integer number; then the monic Gegenbauer polynomials
{C(−N+1=2)
n }n¿0 satisfy (i)
C(−N+1=2)n
n (−1) = (−1)
nC(−N+1=2)
n (1);
(ii)
C(−N+1=2)
n (1) =
−2N+n
−2N + 2n−1
C(−N+1=2)
n−1 (1); n¿1;
(iii)
C(−N+1=2)
n (1) =
(−2N + 1)n
2n(−N+ 1 2)n
; n¿0;
(iv)
C(−N+1=2)
n (1) =C
(−N+1=2)
n (−1) = 0 for n¿2N;
(v)
DkC(−N+1=2)
n (1) =
n! (n−k)!
(−2N + 2k+ 1)n−k
2n−k(−N +k+1 2)n−k
for n¿k;
(vi)
DkC(−N+1=2)
n (1) =D
kC(−N+1=2)
n (−1) = 0 for 06k6N; n¿2N;
(vii)
C(−N+1=2)
2N (x) = (x2−1)N and
(viii) for n¿2N; the following identity holds: C(−N+1=2)
n (x) = (x
2−1)NC(N+1=2)
3. Sobolev orthogonality for {C(−N+1=2)
n }n¿0
Let N¿1 be a given integer number. Let A be a symmetric and positive denite matrix of order 2N. The expression
(f; g)S = (F(1)|F(−1))A(G(1)|G(−1))T+
Z 1
−1
f(2N)(x)g(2N)(x)(1−x2)Ndx; (11)
where
(F(1)|F(−1)) = (f(1); f′
(1); : : : ; f(N−1)(1); f(−1); f′
(−1); : : : ; f(N−1)(−1));
denes an inner product on the linear space of polynomials P. Our aim in this section is to show that it is possible to nd a matrix A such that the sequence of Gegenbauer polynomials {C(−N+1=2)
n }n¿0 is orthogonal with respect to the Sobolev inner product (11).
Proposition 3. LetN¿1 be a given integer number. There exists a symmetric and positive-denite
matrix A such that the sequence {C(−N+1=2)
n }n¿0 is orthogonal with respect to the Sobolev inner
product dened by (11).
Proof. From Proposition 2(iv), we have
C(−N+1=2)
n (1) =C
(−N+1=2)
n (−1) = 0
for n¿2N. On the other hand, from the dierentiation property (9), we deduce that
D2NC(−N+1=2)
n (x) = n! (n−2N)!C
(N+1=2)
n−2N (x)
for n¿2N. Therefore, the polynomial C(−N+1=2)
n , for n¿2N, is orthogonal to the linear space Pn−1 with respect to a Sobolev inner product dened by means of (11), for every symmetric and positive-denite matrix A.
In this way, it only remains to construct a symmetric and positive denite matrix A providing orthogonality of the rst 2N Gegenbauer polynomials {C(−N+1=2)
n }n=0; :::;2N−1. With this aim we are going to construct a regular matrix of order 2N, that we will denote by Q, whose elements are
obtained from the values of the derivatives of the Gegenbauer polynomials at the points −1 and 1. Namely,
Q= (Q(1)|Q(−1))2N×2N;
where
Q(1) = (DkC(−N+1=2)
n (1))n=0; :::;2N−1; k=0; :::; N−1
;
Q(−1) = (DkC(−N+1=2)
n (−1))n=0; :::;2N−1
The matrix Q is regular since it can be expressed as a product of two regular matrices. Indeed, let
us expand the polynomials {C(−N+1=2)
n }n¿0 in powers of x
C(−N+1=2)
n (x) = n
X
j=0
pn; jxj; pn; n6= 0; ∀n¿0: (12)
Let P be the regular matrix constructed with the coecientspn; j, for 06j6n and 06n62N−1
P=
p0;0 0 : : : 0
p1;0 p1;1 : : : 0
..
. ... . .. ...
p2N−1;0 p2N−1;1 : : : p2N−1;2N−1
:
Let us denote by H(x) the matrix obtained from the rst N derivatives of the elements of the Hamel basis of the linear space P2N−1, that is
H(x) = (Dk(xj))j=0; :::;2N−1
k=0; :::; N−1 ;
and denote by V the square matrix obtained from H(x) as follows:
V= (H(1)|H(−1)):
V is a generalized Vandermonde matrix and, therefore, regular. From Eq. (12) we deduce that
Q=P:V, and therefore we conclude the regularity of Q.
For a given diagonal matrix, D, of order 2N and positive elements in the diagonal, we dene the
matrix A of the inner product (11) as follows:
A=Q−1D(Q−1)T:
As we can easily check, A is a positive denite and symmetric matrix and the sequence of polyno-mials {C(−N+1=2)
n }n¿0 is orthogonal with respect to the inner product (11).
4. The linear operator F
In this section, we will consider a linear operator denoted by F, dened on the space of real
polynomials P, which is symmetric with respect to the Sobolev inner product dened in (11). From this operator, we can deduce the existence of several relationships between the sequences of Gegenbauer polynomials {C(−N+1=2)
n }n¿0 and {Cn(N+1=2)}n¿0.
We dene the linear operator F by means of the following expression:
F= (1−x2)ND2N[(1−x2)ND2N]; (13)
where D denotes the derivative operator. In this way, F is a 4Nth-order dierential operator that
Our next result shows that operator F allows us to obtain a representation of the
nondiago-nal Sobolev inner product (11) in terms of the inner product associated with the weight function (N)(x) = (1−x2)N:
Proposition 4. Let f and g be two arbitrary polynomials; then
((1−x2)2Nf; g)
S=
Z 1
−1
f(x)Fg(x)(1−x2)Ndx:
Proof. Since 1 and −1 are zeros of multiplicity N of the polynomial (1−x2)N, we have
((1−x2)2Nf; g)S=
Z 1
−1
D2N[(1−x2)2Nf(x)]D2Ng(x)(1−x2)Ndx:
Integrating by parts 2N times, we deduce
((1−x2)2Nf; g)
S=
Z 1
−1
(1−x2)2Nf(x)D2N[(1−x2)ND2Ng(x)] dx
=
Z 1
−1
f(x)Fg(x)(1−x2)Ndx:
Theorem 5. The linear operator F is symmetric with respect to the Sobolev inner product (11),
i.e.;
(Ff; g)S= (f;Fg)S:
Proof. From the denition of the Sobolev inner product we get
(Ff; g)S=
Z 1
−1
D2N[Ff(x)]D2Ng(x)(1−x2)Ndx
=
Z 1
−1
D2N[(1−x2)ND2N[(1−x2)ND2Nf(x)]]D2Ng(x)(1−x2)Ndx:
Integrating by parts 2N times, we obtain
(Ff; g)
S=
Z 1
−1
(1−x2)ND2N[(1−x2)ND2Nf(x)]D2N[(1−x2)ND2Ng(x)] dx:
As we can see, this last expression is symmetric in f and g, and the result follows interchanging their roles.
A direct computation shows that the linear operator F preserves the degree of the polynomials,
Proposition 6. For n¿2N
As a consequence of Proposition 6, we can deduce that the monic Gegenbauer polynomials C(−N+1=2)
n are the eigenfunctions of the linear operator F.
Proposition 7. For n¿2N; we have
n is a polynomial of exact degree n, it can be expressed as a linear
combi-nation of the polynomials {C(−N+1=2)
The coecients n; i can be obtained from Theorem 5 in the following way:
n; i=
Finally, from the orthogonality of the polynomial C(−N+1=2)
n , we deduce that n; i= 0 for i¡n, and
the result follows by simple inspection of the leading coecient on both sides of (14).
Next, we can deduce some interesting relations involving Gegenbauer polynomials.
Proposition 8. The following relations hold:
(i) (1−x2)2NCn(N+1=2)(x) =
n (x) in terms of the Sobolev orthogonal
From Proposition 4, we can deduce the value of the coecients n; i
and from the orthogonality of the polynomial C(N+1=2)
n , we conclude that n; i= 0, for 06i6n−1.
(ii) If we expand the polynomial FC(−N+1=2)
n in terms of the Gegenbauer polynomials
{Ci(N+1=2)}i∈N, we get
Again, the coecients n; i can be computed from Proposition 4 as follows:
n; i=
and from the orthogonality of the polynomial C(−N+1=2)
n , with respect to the Sobolev inner product,
we conclude that n; i= 0 for 06i¡n−4N.
References
[1] M. Alfaro, M.L. Rezola, T.E. Perez, M.A. Pi˜nar, Sobolev orthogonal polynomials: the discrete–continuous case, submitted.
[2] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [3] K.H. Kwon, L.L. Littlejohn, The Orthogonality of the Laguerre polynomials{L(−k)
n (x)}for positive integerk, Ann.
Numer. Math. 2 (1995) 289–304.
[4] K.H. Kwon, L.L. Littlejohn, Sobolev orthogonal polynomials and second-order dierential equations, Rocky Mt. J. Math., to appear.
[5] T.E. Perez, M.A. Pi˜nar, On Sobolev orthogonality for the generalized Laguerre polynomials, J. Approx. Theory 86 (1996) 278 –285.
[6] T.E. Perez, M.A. Pi˜nar, Sobolev orthogonality and properties of the generalized Laguerre polynomials, in: W.B. Jones, A. Sri Ranga (Eds.), Orthogonal Functions, Moment Theory and Continued Fractions: Theory and Applications 18, Marcel Dekker, New York, 1998, pp. 375–385.
