Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol100.Issue1.1998:
Journal of Computational and Applied Mathematics 100 (1998) 111–120
Sobolev orthogonality for the Gegenbauer
polynomials {Cn(−N +1=2)}n¿0
Mara Alvarez
de Morales a; 1 , Teresa E. Perez b; 2 , Miguel A. Pinar
˜ 2; b; ∗
a
b
Departamento de Matematica Aplicada, Universidad de Granada, 18071 Granada, Spain
Departamento 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 {Cn(−N +1=2) }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
c 1998 Elsevier Science B.V. All rights reserved.
can obtain linear relations with the classical Gegenbauer polynomials.
Keywords: Gegenbauer polynomials; Nondiagonal Sobolev inner product; Sobolev orthogonal polynomials
1. Introduction
For any value of the parameter ¿− 12 monic Gegenbauer polynomials {Cn() }n¿0 can be de ned
by their explicit representation (see [7, p. 84])
[n=2]
Cn() (x) =
2n
X (−1) m (n − m + )
n!
(2x) n−2m ;
(n + ) m=0
m!(n − 2m)!
1
¿− :
2
Simplifying the values of the Gamma function in this expression we get
[n=2]
Cn() (x) =
(−1) m
n! X
(2x) n−2m ;
2n m=0 (n + − m)m m!(n − 2m)!
∗
(1)
Corresponding author. E-mail: [email protected].
Partially supported by Junta de Andaluca, Grupo de Investigacion FQM 0229.
2
Partially supported by Junta de Andaluca, Grupo de Investigacion FQM 0229, DGES under grant PB 95-1205 and
INTAS-93-0219-ext.
1
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 2 4 - 1
112
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
where (n + − m)m denotes the usual Pochhammer symbol de ned 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 de ne Gegenbauer polynomials for
6= 0; −1; −2; : : : .
From the explicit representation (1) we can deduce that the monic Gegenbauer polynomials
{Cn() }n¿0 satisfy, for every real value of 6= 0; −1; −2; : : : ; the three-term recurrence relation
()
C−1
(x) = 0;
C0() (x) = 1;
()
()
(x) +
n() Cn−1
(x);
xCn() (x) = Cn+1
(2)
n¿0;
where
n() =
n(n + 2 − 1)
:
4(n + )(n + − 1)
Whenever 6= − 12 ; − 32 ; : : : ; we have
n() 6= 0 for all n¿1, and Favard’s Theorem (see [2, p. 21])
ensures that the sequence {Cn() }n¿0 is orthogonal with respect to a quasi-de nite linear functional.
Besides, if − 12 ¡ the functional is positive de nite and the polynomials are orthogonal with respect
to the weight (1 − x2 )−1=2 on the interval [−1; 1]. For = − 21 ; − 23 ; : : : ; since
n() vanishes for some
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 polynomials {Cn(−N +1=2) }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 di erent 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 {Ln(−k) }n¿0 ; k¿1, with respect to a Sobolev inner product of the
form
g(0)
g′ (0) Z +∞
+
hf; gi = (f(0); f′ (0); : : : ; f(k−1) (0))A
f(k) (x)g(k) (x)e−x dx
..
0
.
(k−1)
(0)
g
with A a symmetric k × k real matrix. In [4], the same authors showed that Jacobi polynomials
{Pn(−1;−1) }n¿0 , are orthogonal with respect to the inner product
(f; g)1 = d1 f(1)g(1) + d2 f(−1)g(−1) +
Z
1
−1
where d1 and d2 are real numbers.
f′ (x)g′ (x) dx;
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
113
Later, in [5], Perez and Pi˜nar gave a uni ed approach to the orthogonality of the generalized
Laguerre polynomials {Ln( ) }n¿0 , for any real value of the parameter by proving their orthogonality with respect to a Sobolev nondiagonal inner product. In [6], they showed how to use this
orthogonality to obtain di erent 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 de ned by
g(c)
g′ (c)
+ hu; f(N ) g(N ) i;
BS(N ) (f; g) = (f(c); f′ (c); : : : ; f(N −1) (c))A
..
.
g(N −1) (c)
(3)
where u is a quasi-de nite 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
{Pn(−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 de ned from the weight function (0; +N ) (x) = (1 + x) +N ,
c = 1 and A = Q −1 D(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 representation of the monic Gegenbauer polynomials, {Cn() }n¿0 , for ∈ R; 6= 0; −1; −2; : : : we deduce
some of their usual properties, namely the three-term recurrence relation, the di erentiation property,
the second order di erential equation, : : : .
Next, in Section 3, we show that the Gegenbauer polynomials {Cn(−N +1=2) }n¿0 , are orthogonal with
respect to the Sobolev inner product de ned by
g(1)
g′ (1)
..
.
(N −1)
g
(1)
′
(N −1)
′
(N −1)
(f; g)S = f(1); f (1); : : : ; f
(1); f(−1); f (−1); : : : ; f
(−1) A
g(−1)
g′ (−1)
..
.
(N −1)
(−1)
g
+
Z
1
f(2N ) (x)g(2N ) (x)(1 − x2 )N dx;
(4)
−1
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 di erential operator, F, which
is de ned 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 {Cn(−N +1=2) }n¿0 and the classical Gegenbauer polynomials {Cn(N +1=2) }n¿0 .
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M. Alvarez
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2. The Gegenbauer polynomials
For a given real number ¿− 12 , the explicit representation of the nth monic Gegenbauer polynomial is given by
[n=2]
Cn() (x) =
2n
X (−1)m (n − m + )
n!
(2x)n−2m ;
(n + ) m=0
m!(n − 2m)!
n¿0;
(5)
(see [7, p. 84]). As it is well known, for ¿− 12 the sequence {Cn() }n¿0 constitutes a monic orthogonal polynomial sequence associated with the weight function (x) = (1 − x2 )−1=2 on the interval
[−1; 1].
After simpli cation, expression (5) reads
[n=2]
Cn() (x) =
(−1)m
n! X
(2x)n−2m ;
2n m = 0 ( + n − m)m m!(n − 2m)!
n¿0:
(6)
We can observe that for every value of the parameter 6= −l; l = 0; 1; 2; : : : ; expression (6) de nes
a monic polynomial of exact degree n. In this way, for 6= −l; l = 0; 1; 2; : : : we can de ne a family
of monic polynomials {Cn() }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 algebraic 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; for l = 0; 1; 2; : : : . Then; the Gegenbauer polynomials {Cn() }n¿0 ; satisfy the following properties:
(i) Symmetry
Cn() (−x) = (−1)n Cn() (x):
(7)
(ii) Three-term recurrence relation
()
C−1
(x) = 0;
C0() (x) = 1;
()
()
(x) +
n() Cn−1
(x);
xCn() (x) = Cn+1
n¿0;
(8)
where
n() =
n(n + 2 − 1)
:
4(n + )(n + − 1)
(iii) Di erentiation property
(+1)
DCn() (x) = nCn−1
(x);
where D = d=dx:
(9)
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
115
(iv) Second-order di erential equation
(1 − x2 )y′′ − (2 + 1)xy′ + n(n + 2)y = 0;
where y = Cn() (x):
We can observe that the parameter
n() in the three-term recurrence relation does not vanish for
every value of 6= −N + 21 , for N = 1; 2; 3; : : : . Thus, from Favard’s Theorem (see [2, p. 21])
we deduce that the generalized Gegenbauer polynomials {Cn() }n¿0 , 6= −N + 21 for N = 1; 2; 3; : : : ;
are orthogonal with respect to a certain moment functional. For ¿− 12 this moment functional
(−N +1=2)
is positive de nite. However, since
2N
= 0, no orthogonality properties can be deduced from
Favard’s Theorem for the sequence of polynomials {Cn(−N +1=2) }n¿0 for N = 1; 2; 3; : : : .
In order to obtain orthogonality properties for the Gegenbauer polynomials {Cn(−N +1=2) }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
{Cn(−N +1=2) }n¿0 satisfy
(i)
Cn(−N +1=2)n (−1) = (−1)n Cn(−N +1=2) (1);
(ii)
Cn(−N +1=2) (1) =
−2N + n
(−N +1=2)
Cn−1
(1);
−2N + 2n − 1
n¿1;
(iii)
Cn(−N +1=2) (1) =
(−2N + 1)n
;
2n (−N + 21 )n
n¿0;
(iv)
Cn(−N +1=2) (1) = Cn(−N +1=2) (−1) = 0 for n¿2N;
(v)
Dk Cn(−N +1=2) (1) =
(−2N + 2k + 1)n−k
n!
(n − k)! 2n−k (−N + k + 21 )n−k
for n¿k;
(vi)
Dk Cn(−N +1=2) (1) = Dk Cn(−N +1=2) (−1) = 0 for 06k6N; n¿2N;
(vii)
(−N +1=2)
C2N
(x) = (x2 − 1) N and
(viii) for n¿2N; the following identity holds:
(N +1=2)
(x):
Cn(−N +1=2) (x) = (x2 − 1)N Cn−2N
(10)
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M. Alvarez
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3. Sobolev orthogonality for {Cn(−N +1=2) }n¿0
Let N ¿1 be a given integer number. Let A be a symmetric and positive de nite matrix of
order 2N . The expression
T
(f; g)S = (F(1)|F(−1)) A (G(1)|G(−1)) +
Z
1
f(2N ) (x)g(2N ) (x)(1 − x2 ) N dx;
(11)
−1
where
(F(1)|F(−1)) = (f(1); f′ (1); : : : ; f (N −1) (1); f(−1); f′ (−1); : : : ; f (N −1) (−1));
de nes 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 {Cn(−N +1=2) }n¿0
is orthogonal with respect to the Sobolev inner product (11).
Proposition 3. Let N ¿1 be a given integer number. There exists a symmetric and positive-de nite
matrix A such that the sequence {Cn(−N +1=2) }n¿0 is orthogonal with respect to the Sobolev inner
product de ned by (11).
Proof. From Proposition 2(iv), we have
Cn(−N +1=2) (1) = Cn(−N +1=2) (−1) = 0
for n¿2N . On the other hand, from the di erentiation property (9), we deduce that
D 2N Cn(−N +1=2) (x) =
n!
C (N +1=2) (x)
(n − 2N )! n−2N
for n¿2N . Therefore, the polynomial Cn(−N +1=2) , for n¿2N , is orthogonal to the linear space Pn−1
with respect to a Sobolev inner product de ned by means of (11), for every symmetric and positivede nite matrix A.
In this way, it only remains to construct a symmetric and positive de nite matrix A providing
orthogonality of the rst 2N Gegenbauer polynomials {Cn(−N +1=2) }n=0; :::; 2 N −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) = (D k Cn(−N +1=2) (1)) n=0; :::; 2 N −1; ;
k=0; :::; N −1
Q(−1) = (D k Cn(−N +1=2) (−1)) n=0; :::; 2 N −1 :
k=0; :::; N −1
M. Alvarez
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117
The matrix Q is regular since it can be expressed as a product of two regular matrices. Indeed, let
us expand the polynomials {Cn(−N +1=2) }n¿0 in powers of x
Cn(−N +1=2) (x) =
n
X
pn; j x j ;
pn; n 6= 0; ∀n¿0:
(12)
j=0
Let P be the regular matrix constructed with the coecients pn; j , for 06j6n and 06n62N − 1
0
:::
0
p1; 1
:::
0
..
.
..
..
.
p2N −1; 1
:::
p0; 0
p1; 0
P=
..
.
p2N −1; 0
.
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) = (D k (x j )) 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 de ne the
matrix A of the inner product (11) as follows:
A = Q−1 D(Q−1 )T :
As we can easily check, A is a positive de nite and symmetric matrix and the sequence of polynomials {Cn(−N +1=2) }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, de ned on the space of real
polynomials P, which is symmetric with respect to the Sobolev inner product de ned in (11).
From this operator, we can deduce the existence of several relationships between the sequences of
Gegenbauer polynomials {Cn(−N +1=2) }n¿0 and {Cn(N +1=2) }n¿0 .
We de ne the linear operator F by means of the following expression:
F = (1 − x 2 ) N D 2N [(1 − x 2 ) N D 2N ];
(13)
where D denotes the derivative operator. In this way, F is a 4N th-order di erential operator that
vanishes for every polynomial of degree less than or equal to 2N − 1.
118
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
Our next result shows that operator F allows us to obtain a representation of the nondiagonal Sobolev inner product (11) in terms of the inner product associated with the weight function
(N ) (x) = (1 − x 2 ) N :
Proposition 4. Let f and g be two arbitrary polynomials; then
((1 − x 2 ) 2N f; g)S =
Z
1
f(x)Fg(x)(1 − x 2 ) N dx:
−1
Proof. Since 1 and −1 are zeros of multiplicity N of the polynomial (1 − x 2 ) N , we have
2 2N
((1 − x ) f; g)S =
Z
1
D 2N [(1 − x 2 ) 2N f(x)]D 2N g(x)(1 − x 2 ) N dx:
−1
Integrating by parts 2N times, we deduce
2 2N
((1 − x ) f; g)S =
Z
1
Z
1
(1 − x 2 ) 2N f(x)D 2N [(1 − x 2 ) N D 2N g(x)] dx
−1
=
f(x)Fg(x)(1 − x 2 ) N dx:
−1
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 de nition of the Sobolev inner product we get
(Ff; g)S =
Z
1
Z
1
D 2N [Ff(x)]D 2N g(x)(1 − x 2 ) N dx
−1
=
D 2N [(1 − x 2 ) N D 2N [(1 − x 2 ) N D 2N f(x)]]D 2N g(x)(1 − x 2 ) N dx:
−1
Integrating by parts 2N times, we obtain
(Ff; g)S =
Z
1
(1 − x 2 ) N D 2N [(1 − x 2 ) N D 2N f(x)]D 2N [(1 − x 2 ) N D 2N g(x)] dx:
−1
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,
in fact we have
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
119
Proposition 6. For n¿2N
n
Fx =
n!
(n − 2N )!
2
x n + lower degree terms:
As a consequence of Proposition 6, we can deduce that the monic Gegenbauer polynomials
Cn(−N +1=2) are the eigenfunctions of the linear operator F.
Proposition 7. For n¿2N; we have
FCn(−N +1=2) (x) =
n!
(n − 2N )!
2
Cn(−N +1=2) (x):
Proof. Since FCn(−N +1=2) is a polynomial of exact degree n, it can be expressed as a linear combination of the polynomials {Ci(−N +1=2) }i=0 ;:::; n , that means
FCn(−N +1=2) (x) =
n
X
n; i Ci(−N +1=2) (x):
(14)
i=0
The coecients
n; i can be obtained from Theorem 5 in the following way:
n; i =
(FCn(−N +1=2) ; Ci(−N +1=2) )S (Cn(−N +1=2) ; FCi(−N +1=2) )S
=
:
(Ci(−N +1=2) ; Ci(−N +1=2) )S
(Ci(−N +1=2) ; Ci(−N +1=2) )S
Finally, from the orthogonality of the polynomial Cn(−N +1=2) , 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 − x 2 ) 2N Cn(N +1=2) (x) =
n+4N
X
n; i Ci(−N +1=2) (x);
n¿0;
(15)
i=n
(ii)
FCn(−N +1=2) (x) =
n
X
n; i Ci(N +1=2) (x);
n¿4N:
(16)
i=n−4N
Proof. (i) If we expand the polynomial (1 − x 2 ) 2N Cn(N +1=2) (x) in terms of the Sobolev orthogonal
polynomials {Ci(−N +1=2) }i¿0 , we get
(1 − x 2 ) 2N Cn(N +1=2) (x) =
n+4N
X
i=0
n; i Ci(−N +1=2) (x):
120
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
From Proposition 4, we can deduce the value of the coecients n; i
n; i =
=
((1 − x 2 ) 2N Cn(N +1=2) ; Ci(−N +1=2) )S
(Ci(−N +1=2) ; Ci(−N +1=2) )S
R1
−1
Cn(N +1=2) (x)F(Ci(−N +1=2) (x))(1 − x 2 ) N dx
(Ci(−N +1=2) ; Ci(−N +1=2) )S
;
and from the orthogonality of the polynomial Cn(N +1=2) , we conclude that n; i = 0, for 06i6n − 1.
(ii) If we expand the polynomial FCn(−N +1=2) in terms of the Gegenbauer polynomials
{Ci(N +1=2) }i∈N , we get
FCn(−N +1=2) (x) =
n
X
n; i Ci(N +1=2) (x):
i=0
Again, the coecients n; i can be computed from Proposition 4 as follows:
n; i =
R1
−1
Ci(N +1=2) (x)FCn(−N +1=2) (x)(1 − x 2 ) N dx
R1
−1
= R1
−1
Ci(N +1=2) (x)Ci(N +1=2) (x)(1 − x 2 ) N dx
((1 − x 2 ) 2N Ci(N +1=2) ; Cn(−N +1=2) )S
Ci(N +1=2) (x)Ci(N +1=2) (x)(1 − x 2 ) N dx
and from the orthogonality of the polynomial Cn(−N +1=2) , 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 {Ln(−k) (x)} for positive integer k, Ann.
Numer. Math. 2 (1995) 289–304.
[4] K.H. Kwon, L.L. Littlejohn, Sobolev orthogonal polynomials and second-order di erential 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.
[7] G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.
Sobolev orthogonality for the Gegenbauer
polynomials {Cn(−N +1=2)}n¿0
Mara Alvarez
de Morales a; 1 , Teresa E. Perez b; 2 , Miguel A. Pinar
˜ 2; b; ∗
a
b
Departamento de Matematica Aplicada, Universidad de Granada, 18071 Granada, Spain
Departamento 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 {Cn(−N +1=2) }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
c 1998 Elsevier Science B.V. All rights reserved.
can obtain linear relations with the classical Gegenbauer polynomials.
Keywords: Gegenbauer polynomials; Nondiagonal Sobolev inner product; Sobolev orthogonal polynomials
1. Introduction
For any value of the parameter ¿− 12 monic Gegenbauer polynomials {Cn() }n¿0 can be de ned
by their explicit representation (see [7, p. 84])
[n=2]
Cn() (x) =
2n
X (−1) m (n − m + )
n!
(2x) n−2m ;
(n + ) m=0
m!(n − 2m)!
1
¿− :
2
Simplifying the values of the Gamma function in this expression we get
[n=2]
Cn() (x) =
(−1) m
n! X
(2x) n−2m ;
2n m=0 (n + − m)m m!(n − 2m)!
∗
(1)
Corresponding author. E-mail: [email protected].
Partially supported by Junta de Andaluca, Grupo de Investigacion FQM 0229.
2
Partially supported by Junta de Andaluca, Grupo de Investigacion FQM 0229, DGES under grant PB 95-1205 and
INTAS-93-0219-ext.
1
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 2 4 - 1
112
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
where (n + − m)m denotes the usual Pochhammer symbol de ned 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 de ne Gegenbauer polynomials for
6= 0; −1; −2; : : : .
From the explicit representation (1) we can deduce that the monic Gegenbauer polynomials
{Cn() }n¿0 satisfy, for every real value of 6= 0; −1; −2; : : : ; the three-term recurrence relation
()
C−1
(x) = 0;
C0() (x) = 1;
()
()
(x) +
n() Cn−1
(x);
xCn() (x) = Cn+1
(2)
n¿0;
where
n() =
n(n + 2 − 1)
:
4(n + )(n + − 1)
Whenever 6= − 12 ; − 32 ; : : : ; we have
n() 6= 0 for all n¿1, and Favard’s Theorem (see [2, p. 21])
ensures that the sequence {Cn() }n¿0 is orthogonal with respect to a quasi-de nite linear functional.
Besides, if − 12 ¡ the functional is positive de nite and the polynomials are orthogonal with respect
to the weight (1 − x2 )−1=2 on the interval [−1; 1]. For = − 21 ; − 23 ; : : : ; since
n() vanishes for some
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 polynomials {Cn(−N +1=2) }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 di erent 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 {Ln(−k) }n¿0 ; k¿1, with respect to a Sobolev inner product of the
form
g(0)
g′ (0) Z +∞
+
hf; gi = (f(0); f′ (0); : : : ; f(k−1) (0))A
f(k) (x)g(k) (x)e−x dx
..
0
.
(k−1)
(0)
g
with A a symmetric k × k real matrix. In [4], the same authors showed that Jacobi polynomials
{Pn(−1;−1) }n¿0 , are orthogonal with respect to the inner product
(f; g)1 = d1 f(1)g(1) + d2 f(−1)g(−1) +
Z
1
−1
where d1 and d2 are real numbers.
f′ (x)g′ (x) dx;
M. Alvarez
de Morales et al. / Journal of Computational and Applied Mathematics 100 (1998) 111–120
113
Later, in [5], Perez and Pi˜nar gave a uni ed approach to the orthogonality of the generalized
Laguerre polynomials {Ln( ) }n¿0 , for any real value of the parameter by proving their orthogonality with respect to a Sobolev nondiagonal inner product. In [6], they showed how to use this
orthogonality to obtain di erent 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 de ned by
g(c)
g′ (c)
+ hu; f(N ) g(N ) i;
BS(N ) (f; g) = (f(c); f′ (c); : : : ; f(N −1) (c))A
..
.
g(N −1) (c)
(3)
where u is a quasi-de nite 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
{Pn(−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 de ned from the weight function (0; +N ) (x) = (1 + x) +N ,
c = 1 and A = Q −1 D(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 representation of the monic Gegenbauer polynomials, {Cn() }n¿0 , for ∈ R; 6= 0; −1; −2; : : : we deduce
some of their usual properties, namely the three-term recurrence relation, the di erentiation property,
the second order di erential equation, : : : .
Next, in Section 3, we show that the Gegenbauer polynomials {Cn(−N +1=2) }n¿0 , are orthogonal with
respect to the Sobolev inner product de ned by
g(1)
g′ (1)
..
.
(N −1)
g
(1)
′
(N −1)
′
(N −1)
(f; g)S = f(1); f (1); : : : ; f
(1); f(−1); f (−1); : : : ; f
(−1) A
g(−1)
g′ (−1)
..
.
(N −1)
(−1)
g
+
Z
1
f(2N ) (x)g(2N ) (x)(1 − x2 )N dx;
(4)
−1
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 di erential operator, F, which
is de ned 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 {Cn(−N +1=2) }n¿0 and the classical Gegenbauer polynomials {Cn(N +1=2) }n¿0 .
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2. The Gegenbauer polynomials
For a given real number ¿− 12 , the explicit representation of the nth monic Gegenbauer polynomial is given by
[n=2]
Cn() (x) =
2n
X (−1)m (n − m + )
n!
(2x)n−2m ;
(n + ) m=0
m!(n − 2m)!
n¿0;
(5)
(see [7, p. 84]). As it is well known, for ¿− 12 the sequence {Cn() }n¿0 constitutes a monic orthogonal polynomial sequence associated with the weight function (x) = (1 − x2 )−1=2 on the interval
[−1; 1].
After simpli cation, expression (5) reads
[n=2]
Cn() (x) =
(−1)m
n! X
(2x)n−2m ;
2n m = 0 ( + n − m)m m!(n − 2m)!
n¿0:
(6)
We can observe that for every value of the parameter 6= −l; l = 0; 1; 2; : : : ; expression (6) de nes
a monic polynomial of exact degree n. In this way, for 6= −l; l = 0; 1; 2; : : : we can de ne a family
of monic polynomials {Cn() }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 algebraic 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; for l = 0; 1; 2; : : : . Then; the Gegenbauer polynomials {Cn() }n¿0 ; satisfy the following properties:
(i) Symmetry
Cn() (−x) = (−1)n Cn() (x):
(7)
(ii) Three-term recurrence relation
()
C−1
(x) = 0;
C0() (x) = 1;
()
()
(x) +
n() Cn−1
(x);
xCn() (x) = Cn+1
n¿0;
(8)
where
n() =
n(n + 2 − 1)
:
4(n + )(n + − 1)
(iii) Di erentiation property
(+1)
DCn() (x) = nCn−1
(x);
where D = d=dx:
(9)
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115
(iv) Second-order di erential equation
(1 − x2 )y′′ − (2 + 1)xy′ + n(n + 2)y = 0;
where y = Cn() (x):
We can observe that the parameter
n() in the three-term recurrence relation does not vanish for
every value of 6= −N + 21 , for N = 1; 2; 3; : : : . Thus, from Favard’s Theorem (see [2, p. 21])
we deduce that the generalized Gegenbauer polynomials {Cn() }n¿0 , 6= −N + 21 for N = 1; 2; 3; : : : ;
are orthogonal with respect to a certain moment functional. For ¿− 12 this moment functional
(−N +1=2)
is positive de nite. However, since
2N
= 0, no orthogonality properties can be deduced from
Favard’s Theorem for the sequence of polynomials {Cn(−N +1=2) }n¿0 for N = 1; 2; 3; : : : .
In order to obtain orthogonality properties for the Gegenbauer polynomials {Cn(−N +1=2) }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
{Cn(−N +1=2) }n¿0 satisfy
(i)
Cn(−N +1=2)n (−1) = (−1)n Cn(−N +1=2) (1);
(ii)
Cn(−N +1=2) (1) =
−2N + n
(−N +1=2)
Cn−1
(1);
−2N + 2n − 1
n¿1;
(iii)
Cn(−N +1=2) (1) =
(−2N + 1)n
;
2n (−N + 21 )n
n¿0;
(iv)
Cn(−N +1=2) (1) = Cn(−N +1=2) (−1) = 0 for n¿2N;
(v)
Dk Cn(−N +1=2) (1) =
(−2N + 2k + 1)n−k
n!
(n − k)! 2n−k (−N + k + 21 )n−k
for n¿k;
(vi)
Dk Cn(−N +1=2) (1) = Dk Cn(−N +1=2) (−1) = 0 for 06k6N; n¿2N;
(vii)
(−N +1=2)
C2N
(x) = (x2 − 1) N and
(viii) for n¿2N; the following identity holds:
(N +1=2)
(x):
Cn(−N +1=2) (x) = (x2 − 1)N Cn−2N
(10)
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3. Sobolev orthogonality for {Cn(−N +1=2) }n¿0
Let N ¿1 be a given integer number. Let A be a symmetric and positive de nite matrix of
order 2N . The expression
T
(f; g)S = (F(1)|F(−1)) A (G(1)|G(−1)) +
Z
1
f(2N ) (x)g(2N ) (x)(1 − x2 ) N dx;
(11)
−1
where
(F(1)|F(−1)) = (f(1); f′ (1); : : : ; f (N −1) (1); f(−1); f′ (−1); : : : ; f (N −1) (−1));
de nes 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 {Cn(−N +1=2) }n¿0
is orthogonal with respect to the Sobolev inner product (11).
Proposition 3. Let N ¿1 be a given integer number. There exists a symmetric and positive-de nite
matrix A such that the sequence {Cn(−N +1=2) }n¿0 is orthogonal with respect to the Sobolev inner
product de ned by (11).
Proof. From Proposition 2(iv), we have
Cn(−N +1=2) (1) = Cn(−N +1=2) (−1) = 0
for n¿2N . On the other hand, from the di erentiation property (9), we deduce that
D 2N Cn(−N +1=2) (x) =
n!
C (N +1=2) (x)
(n − 2N )! n−2N
for n¿2N . Therefore, the polynomial Cn(−N +1=2) , for n¿2N , is orthogonal to the linear space Pn−1
with respect to a Sobolev inner product de ned by means of (11), for every symmetric and positivede nite matrix A.
In this way, it only remains to construct a symmetric and positive de nite matrix A providing
orthogonality of the rst 2N Gegenbauer polynomials {Cn(−N +1=2) }n=0; :::; 2 N −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) = (D k Cn(−N +1=2) (1)) n=0; :::; 2 N −1; ;
k=0; :::; N −1
Q(−1) = (D k Cn(−N +1=2) (−1)) n=0; :::; 2 N −1 :
k=0; :::; N −1
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117
The matrix Q is regular since it can be expressed as a product of two regular matrices. Indeed, let
us expand the polynomials {Cn(−N +1=2) }n¿0 in powers of x
Cn(−N +1=2) (x) =
n
X
pn; j x j ;
pn; n 6= 0; ∀n¿0:
(12)
j=0
Let P be the regular matrix constructed with the coecients pn; j , for 06j6n and 06n62N − 1
0
:::
0
p1; 1
:::
0
..
.
..
..
.
p2N −1; 1
:::
p0; 0
p1; 0
P=
..
.
p2N −1; 0
.
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) = (D k (x j )) 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 de ne the
matrix A of the inner product (11) as follows:
A = Q−1 D(Q−1 )T :
As we can easily check, A is a positive de nite and symmetric matrix and the sequence of polynomials {Cn(−N +1=2) }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, de ned on the space of real
polynomials P, which is symmetric with respect to the Sobolev inner product de ned in (11).
From this operator, we can deduce the existence of several relationships between the sequences of
Gegenbauer polynomials {Cn(−N +1=2) }n¿0 and {Cn(N +1=2) }n¿0 .
We de ne the linear operator F by means of the following expression:
F = (1 − x 2 ) N D 2N [(1 − x 2 ) N D 2N ];
(13)
where D denotes the derivative operator. In this way, F is a 4N th-order di erential operator that
vanishes for every polynomial of degree less than or equal to 2N − 1.
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Our next result shows that operator F allows us to obtain a representation of the nondiagonal Sobolev inner product (11) in terms of the inner product associated with the weight function
(N ) (x) = (1 − x 2 ) N :
Proposition 4. Let f and g be two arbitrary polynomials; then
((1 − x 2 ) 2N f; g)S =
Z
1
f(x)Fg(x)(1 − x 2 ) N dx:
−1
Proof. Since 1 and −1 are zeros of multiplicity N of the polynomial (1 − x 2 ) N , we have
2 2N
((1 − x ) f; g)S =
Z
1
D 2N [(1 − x 2 ) 2N f(x)]D 2N g(x)(1 − x 2 ) N dx:
−1
Integrating by parts 2N times, we deduce
2 2N
((1 − x ) f; g)S =
Z
1
Z
1
(1 − x 2 ) 2N f(x)D 2N [(1 − x 2 ) N D 2N g(x)] dx
−1
=
f(x)Fg(x)(1 − x 2 ) N dx:
−1
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 de nition of the Sobolev inner product we get
(Ff; g)S =
Z
1
Z
1
D 2N [Ff(x)]D 2N g(x)(1 − x 2 ) N dx
−1
=
D 2N [(1 − x 2 ) N D 2N [(1 − x 2 ) N D 2N f(x)]]D 2N g(x)(1 − x 2 ) N dx:
−1
Integrating by parts 2N times, we obtain
(Ff; g)S =
Z
1
(1 − x 2 ) N D 2N [(1 − x 2 ) N D 2N f(x)]D 2N [(1 − x 2 ) N D 2N g(x)] dx:
−1
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,
in fact we have
M. Alvarez
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119
Proposition 6. For n¿2N
n
Fx =
n!
(n − 2N )!
2
x n + lower degree terms:
As a consequence of Proposition 6, we can deduce that the monic Gegenbauer polynomials
Cn(−N +1=2) are the eigenfunctions of the linear operator F.
Proposition 7. For n¿2N; we have
FCn(−N +1=2) (x) =
n!
(n − 2N )!
2
Cn(−N +1=2) (x):
Proof. Since FCn(−N +1=2) is a polynomial of exact degree n, it can be expressed as a linear combination of the polynomials {Ci(−N +1=2) }i=0 ;:::; n , that means
FCn(−N +1=2) (x) =
n
X
n; i Ci(−N +1=2) (x):
(14)
i=0
The coecients
n; i can be obtained from Theorem 5 in the following way:
n; i =
(FCn(−N +1=2) ; Ci(−N +1=2) )S (Cn(−N +1=2) ; FCi(−N +1=2) )S
=
:
(Ci(−N +1=2) ; Ci(−N +1=2) )S
(Ci(−N +1=2) ; Ci(−N +1=2) )S
Finally, from the orthogonality of the polynomial Cn(−N +1=2) , 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 − x 2 ) 2N Cn(N +1=2) (x) =
n+4N
X
n; i Ci(−N +1=2) (x);
n¿0;
(15)
i=n
(ii)
FCn(−N +1=2) (x) =
n
X
n; i Ci(N +1=2) (x);
n¿4N:
(16)
i=n−4N
Proof. (i) If we expand the polynomial (1 − x 2 ) 2N Cn(N +1=2) (x) in terms of the Sobolev orthogonal
polynomials {Ci(−N +1=2) }i¿0 , we get
(1 − x 2 ) 2N Cn(N +1=2) (x) =
n+4N
X
i=0
n; i Ci(−N +1=2) (x):
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M. Alvarez
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From Proposition 4, we can deduce the value of the coecients n; i
n; i =
=
((1 − x 2 ) 2N Cn(N +1=2) ; Ci(−N +1=2) )S
(Ci(−N +1=2) ; Ci(−N +1=2) )S
R1
−1
Cn(N +1=2) (x)F(Ci(−N +1=2) (x))(1 − x 2 ) N dx
(Ci(−N +1=2) ; Ci(−N +1=2) )S
;
and from the orthogonality of the polynomial Cn(N +1=2) , we conclude that n; i = 0, for 06i6n − 1.
(ii) If we expand the polynomial FCn(−N +1=2) in terms of the Gegenbauer polynomials
{Ci(N +1=2) }i∈N , we get
FCn(−N +1=2) (x) =
n
X
n; i Ci(N +1=2) (x):
i=0
Again, the coecients n; i can be computed from Proposition 4 as follows:
n; i =
R1
−1
Ci(N +1=2) (x)FCn(−N +1=2) (x)(1 − x 2 ) N dx
R1
−1
= R1
−1
Ci(N +1=2) (x)Ci(N +1=2) (x)(1 − x 2 ) N dx
((1 − x 2 ) 2N Ci(N +1=2) ; Cn(−N +1=2) )S
Ci(N +1=2) (x)Ci(N +1=2) (x)(1 − x 2 ) N dx
and from the orthogonality of the polynomial Cn(−N +1=2) , 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 {Ln(−k) (x)} for positive integer k, Ann.
Numer. Math. 2 (1995) 289–304.
[4] K.H. Kwon, L.L. Littlejohn, Sobolev orthogonal polynomials and second-order di erential 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.
[7] G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.