ETNA
Electronic Transactions on Numerical Analysis.
Volume 24, pp. 55-65, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF
ORDER (II) M. ISABEL BUENO , FRANCISCO MARCELLÁN , AND JORGE SÁNCHEZ-RUIZ Abstract. Given a symmetrized Sobolev inner product of order , the corresponding sequence of monic orthogonal polynomials satisfies , for certain sequences of monic
polynomials and . In this paper we consider the particular case when all the measures that define the
symmetrized Sobolev inner product are equal, absolutely continuous and semiclassical. Under such restrictions, we
give explicit algebraic relations between the sequences and , as well as higher-order recurrence relations
that they satisfy.
Key words. Sobolev inner product, orthogonal polynomials, semiclassical linear functionals, recurrence relation, symmetrization process
AMS subject classification. 42C05
1. Introduction. Let us consider the following inner product defined in the linear space
!
, where denotes the linear space of polynomials with
real
coefficients,
?E
?E
!#"$!
?
23
(1.1)
%'&)(+*-,/.10
A
?GF
23
> ?@BA6C
&4*65879;:=<
FFF
&)D
* D
587
?E
7H9 (I7
(
(I7
In the previous expression
denote positive and
absolutely continuous Borel
?
measures supported on a subset of the real
line
and
such
that
the
corresponding sequences of
<
C
moments
are
finite,
&
denotes
the
J
th
derivative
of
&
,
and
are
nonnegative real numbers
D
C
(
0NM ). The inner product given in (1.1) is known as Sobolev inner product of order
[7].<LSobolev
inner products and their corresponding sequences of orthogonal polynomials
K
have been exhaustively studied during the last ten years, although most of the results have
been obtained for 0PO .
A
The product %IQ(RQ , . is said to be symmetrized if %TSU(ISVW, . 0XM when YZ:\[ is an odd
nonnegative integer. In this case, 7 9 , 7 ,..., 7
are supported on a subset of the real line which
is symmetric with respect to the origin and the
measures themselves are also symmetric, so
<
?E
A
]
that
?
FFF
F
A
2 3
^D
U-_
0
^
S
U-_
587
0`Ma(bJc0`Md(eO8(
(
(fYhgi
This concept extends the definition of symmetric linear functional [5] to the bilinear case.
Assume that %IQ(RQ ,/. is quasi-definite, that is, there exists a sequence jlk
of polynomiUm
als orthogonal with respect to %nQ(eQo,+. . If %nQ(eQo,/. is symmetrized, then there exist another two
F
sequences of polynomials jp
and jq
such thatA
Um
(1.2)
k
^
Ur
Ss0`p
^
Ur
S
Um
st(uk
^
^
U-_
r
Ss0`Svq
UHr
S
s
In the sequel, we will refer to these two sequences as the symmetric components of jwk
.
U m
In this paper we consider the bilinear symmetrization problem associated with (1.1), i.e.
the analog in the bilinear case of Chihara’s linear symmetrization problem [5], which consists
in:x
Received December 15, 2004. Accepted for publication Octuber 31, 2005. Recommended by R. AlvarezNodarse.
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés,
Madrid, Spain (mbueno@math.uc3m.es, pacomarc@ing.uc3m.es, jsanchez@math.uc3m.es).
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain.
55
ETNA
Kent State University
etna@mcs.kent.edu
56
M. ISABEL BUENO, FRANCISCO MARCELLÁN, AND JORGE SÁNCHEZ-RUIZ
and jq
(1) finding the explicit expressions of the bilinear functionals such that jp
Um
Uvm
are the corresponding sequences of orthogonal polynomials,
(2) determining recurrence relations with a finite number of terms that jlp
and jq
U m
U m
satisfy, and
(3) obtaining explicit algebraic relations between both sequences.
The problem (1) was solved for Sobolev inner products of order O in [3], and for general
in [4]. As regards problems (2) and (3), till now they have only been solved in the particular
case when
0XO and the two measures that define the product %nQ(eQo, . are equal, absolutely
Ly
continuous and semiclassical [3]. In this paper we extend these results for an arbitrary
O ,
under the same restrictions on the measures involved.
The structure of the paper is the following: First, in Section 2, we present some auxiliary
results related with semiclassical functionals and semiclassical measures. In Section 3, we
find explicit algebraic relations between the sequences jp
and jq
. In Section 4, we
Um
Um
, jq
determine recurrence relations with a finite number of terms that the sequences jlp
Um
Uvm
and jlk
satisfy. Finally, in Section 5, we apply our general results to a particular case of
Udm
the so-called Freud-Sobolev polynomials [2].
2. Auxiliary results. Consider a quasi-definite linear functional
representation
2 3
(2.1)
z
r
&s0
z
!
in , with integral
2 3
&
SsI57{0
r
&
r
Ss}|
r
SsI5Sh(
where 7 is an absolutely continuous positive Borel measure and | is the corresponding weight
function. The functional z is said to be a semiclassical linear functional if
(2.2)
~
U s0‚ U (
r€
y
y
where and  are polynomials with ƒ„R
denotes the
sW0‡†
M and ƒ„e
1s
O , and ~

r€
r
derivative operator. The condition of being semiclassical can also be characterized in terms
of the weight function | :
P ROPOSITION 2.1. [6] Let z be a semiclassical linear functional with integral representation
(2.1), where | is a continuously differentiable function in an interval ˆ ‰(+Š‹ satisfying
ŒŽl‘“’” •
!
Ss–&
Ss}|
Ss;0—M with &Zg
. Then,
6r
r
r
(2.3)
r˜
|™sIš0`
|(
and | is said to be a semiclassical weight function. Equation (2.3) is the so-called Pearson
equation.
D EFINITION 2.2. [8] Given a semiclassical linear functional z , let ›œ be the set of all
F
the pairs of polynomials satisfying (2.2). Then, the class Ÿ ž of z is defined
as
Ÿ
(2.4)
ž
E}¤-¥H¦¨§
Ž ¡
ӣ
0
Ž©wª
jƒ«„e
D¢
L EMMA 2.3. [7] If
integer ,
r€
sG¬#­®(/ƒ„R
r
1sG¬¯O
m±°
is a semiclassical weight function, then for every nonnegative
|

<
r
Ssn~
|
<²
r
SsI³W0\
r
S)(
s}|
r
Ss(
where


F
S)(+M8s
0´O
(
r
r
S)(
s
0
Br
Ssn™š
r
SB(
¬¯OsH:µ
r
S)(
¬¯Oseˆ 
r
SsG¬

š
r
Ss}‹ (
¶y
O
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etna@mcs.kent.edu
CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF ORDER
57
(II)
L EMMA 2.4. [7] Given a semiclassical weight function | , the polynomials
defined in the previous lemma satisfy
ƒ«„e

S)(
r
²
Ÿ
sI³·
r
ž
Ly
:`Os(
?
> @
(2.5)
¸ D
<
r
9
V
C
> ?@ V
C
<
0
¼Ols V
[
V
V1_

JH¼
9¨»
r
<¾½
Ss¿
r
S)(
s
Ma(
where Ÿ ž is the class of the semiclassical linear functional defined by | .
!
Consider the following linear differential operator in the linear space :
E™¹

V1_
S)(/[X¬ÀJnsn~
r
?
(
9
where 9 0´O and ~
0`Á , the identity operator.
E
P ROPOSITION 2.5. Let z be a semiclassical
linear functional with integral representation (2.1). Using the notations introduced
above, for all nonnegative
integers and Y ,
E
Ž ¡
Ÿ
†®:
ž ¬À†ws
j
(IY
.
(1) If Ÿ ž ¬À† ¯M , then ƒ„R… ¸ D r SvUds/³·ÃYÄ:
r
m
²
(2) If Ÿ ž ¬À† ·¯M , then ƒ„R… ¸ D < SvUds ³ 0\YÄ:
† .
r
²
<
Proof. From (2.5),
E
?
> @
¸
D
r
<
S U s
0
?
C
<
r
9
V
> ?@V
¼Os V
[
V
V1_

J¼
9¨»
?
r
<¾½
Ssn
V1_
S)(I[ŬÀJ¿sI~
r
r
F
S U s
Notice that ~ÆV1_ r SvUdsh0ÇM when [È:ÉJh¶Y . It follows that theŽ¨upper
bounds on the
¡
[
J can be replaced, respectively, by Ê
0
j
(/Y
0
summations
over
and
and [Ë
Ê
Ž¨¡
m
j[h(/Y{µ[
, with the condition that empty sum equals zero if Y—ÌÍ[ . Using Lemma
m
E
2.4,
ƒ„R
¸ D
²
<
r
S U s ³
?@
@ ©wª
Ž
·
jR†
9tÎÐÏ
r
?@ 9ÐVÎ Ï
V Ž
@ ©wª
< jRYÄ:
9tÎÐÏ
9ÐVÎ Ï
0
¬À[Ñ:µJnsH:
Ÿ
†®:µ[
r
ž
r
Ÿ
[Ò¬ÀJ¿s
¬À†-sc¬ÀJ
Ÿ
r
ž
:\OlsH:ÓY¬À[Ò¬ÀJ
m
F
r
ž
:íW¬h†-s
m
V
From (2.4), we know that Ÿ ž
ƒ«„e
¸
²
y
<
E
:¯­4¬À†
D
r
<
S
. Therefore, the maximum is attained
for J60—M ,
F
M
U s ³
@
Ž¨©-ª
·
jY:
9tÎ Ï
V
Ÿ
†a:Ó[
r
ž
¬h†-s
m
<
E ‚M , then the maximum is attained for [Ô0
Ê , and the result in (1) follows. On the
If Ÿ ž ¬À†¨
other hand, if Ÿ ž ¬h†Õ·‚M , then¹ the maximum is attained for [Å0`M ; since the term [Å0`JG0\M
in ¸ D r SUds has exact degree YÄ: Ž©wª † , (2) holds.
Ÿ E
jR†w(wž . Then, for all nonnegative
C< OROLLARY 2.6. Let Ÿ 0
integers (/Y ,
F
m
ƒ„e
¸
²
D
<
r
S U s ³
Ÿ
·ÃYÄ:
3. Explicit algebraic relations between jp
and jq
U m
quasi-definite Sobolev inner product given by (1.1) with 7
?
2 3
(3.1)
%Ö&)(/*-, .
0
> ?@BA
&4*G57\:
<
C
?
U
. Let us consider a symmetrized
for all J ( M ·ÃJ
·
), i.e.,
m
0
‚
7
?E
?E
F
2 3
&HD
* D
587
ETNA
Kent State University
etna@mcs.kent.edu
58
M. ISABEL BUENO, FRANCISCO MARCELLÁN, AND JORGE SÁNCHEZ-RUIZ
the sequence of monic polynomials orthogonal with respect to (3.1), and
We denote by jwk
Um
by jlp
and jlq
the corresponding symmetric components defined by (1.2). The weight
Um
Um
function | satisfying 57{0‚| SsI5S and the corresponding linear functional z given by (2.1)
r
are both symmetric. Furthermore, the sequence jR×
of monic polynomials orthogonal with
U m
respect to z satisfies a three-term recurrence relation,
]
A
(3.2)
Sv×
]
Ss0\×
Ur
A
U-_
SsB:
r
×
U
U
y
Ss(fY
r
O
(
½
where
U
0\M
, and there exists a sequence of monic polynomials
jlØ
A
K
(3.3)
^
^
×
Ur
Ss™0—Ø
Ur
S
^
^
s(f×
such that
U m
U_
Ss0`SØ
r
U
S
r
s(
where jwØ denotes the sequence of monic kernel polynomials associated with jlØ
[5].
U m
U m
In the sequel, we assume that z is a semiclassical linear functional, and we use the
notations introduced in the previous section. The following proposition states an algebraic
relation between the sequences jwk
and j×
that will be useful to determine algebraic
U m
U m
relations between jp
and jlq
.
U m
U m
E
L EMMA 3.1. [7] Let & and * be arbitrary polynomials.
Then,
F
%
Ù
”Ú
U
&B(/*-,/.;0—z

&¸
D
²
<
*-sI³
r
<
P ROPOSITION 3.2. For every nonnegative integer
such that
yÅ
Y
Ÿ
, there exist real numbers
F
@ U-> _
(3.4)

r
<
Ss¿×
UHr
Ss0
Ù
<1Û
Ú
”Ú
k
U
.
U
Ú
r
Ss
½<
get
Proof. Expanding the polynomial

<
r
Ss¿×
Ur
in terms of the Sobolev polynomials we
Ss
F
U-> _ @

r
<
Ss¿×
U
r
Ss0
Ú
”Ú
Ù
<1Û
k
U
9
Ù
U
0
%

<
%˜k
Ú
Sst(+k
S s/, .
UHr
r
Ú
Sst(+k
Ss/, .
r
r
Ss¿×
r Ú
0
z
U
²
×
%˜k
Ss
r
”Ú
Then, we use Lemma 3.1 to compute the coefficients Ù
”Ú
Ú
,
Ur Ú
E
Ssn¸
r
D
Sst(+k
<
Ú ²
k
r
F
Ú
r
Ss ³³
Ss/, .
Since j×
is the sequence of polynomials orthogonal with respect to z , Corollary 2.6 imUm ” Ú
Ÿ
0\M if M¨·#ܐ̯Y$¬
plies that Ù
.
U
R EMARK 3.3. If in the above proof Proposition 2.5 is used instead of Corollary 2.6, a
sharper lower bound can be obtained for the summation in (3.4). This means that the first
terms of the sum in (3.4) may be zero. However, since the use of Proposition 2.5 would require
two parallel developments in what follows, for the sake of brevity we use the general bound
given by Corollary 2.6.
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CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF ORDER
3.1. Semiclassical functional z of even class.
P ROPOSITION 3.4. If the class Ÿ ž of the functional z
algebraic relation between the sequences jp
and jq
U
Ï
@ V;> _
<
Ú
Ù
Û
V
<
Ú]
Ù
V
V
½<
m
A
^
V1_
Ù
^
q
_
AV
½<
Ú
snq
r
r
Ss
Û
Ss
½
Sst(
r
V
<
A
_
Ï.
½
Ï
V1_
” Ú
^
^
V
q
is even, then the following explicit
is obtained,
<1Û
A
:
.
½
^
U
]
_ A
^
”
V1_
” Ú
^
V
Ÿ
A
A
V1_
”
Ù ½^ <
Ù
Ss0
r
^
r A
Ï.
V
Ÿ
p
A
Ûн
^
:
Ú
Ï
½<
@ _ >
V1
:
^
A V
Ï.
(3.5)
” Ú
^
m
59
(II)
½
Ý-­ .
where †“
Ê 0`†-Ý­ and Ê 0
Proof. Since Ÿ ž is an even number,  is an even polynomial [3, Prop. 2.6]. That is, there
F
exists another polynomial ž such that

(3.6)
6r
^
ž S
 r
6
Ss™0
s
Ÿ
Therefore, † and are also even numbers. We write †“0—­
For YZ0—­w[ , Proposition 3.2 reads
(3.7)

Ss0
r
V
Ú
Ÿ
0`­
Ê
.
F
^
@ ;
V > _
^
Ss¿×
r
<
and Ÿ
†Ê
Ù
<;Û
^
”Ú
^
.
V
Ú
k
V
Ss
r
½<
Since the term in the left-hand side of the previous
identity is an even polynomial, we get

^
Ss¿×
r
<
V r
Ss™0
F
Ï
@ V1> _
<
Ú
Ù
Û
” Ú
^
^
V
^
k
V
Ï.
Ú
Ss
r
½<
Taking into account (1.2), (3.3), and (3.6), the previous
equation can be rewritten as
(3.8)
ž

Ss/Ø
r
<
Ss™0
V¨r
<
Ú
V
ž
Ss/Ø
r
<
V
Ss0
r
p
V
Ss
r
A
Ï
<
Ú
^
Ú
½<
@ V1> _

Ù
Û
” Ú
^
Ï.
In a similar way, for YÞ0`­w[N:\O we obtain
(3.9)
F
Ï
@ V1> _
Ù
Û
F
” Ú
^
^
V1_
Ï.
V
A
q
Ú
_
Ss
r
½<
]
Taking into account (3.3) and (3.2) with Y replaced
by ­w[ A , we F get
(3.10)
Ø
Multiplying (3.10) by
<
Ú
(3.11)
V
V
p
Û
Ú
r
^
Ss):
Ø
V
V
r
Ss
Ss
A
A
Ï
<
Ú
” Ú
^
^
½<
@ V1> _
0
r
<
Ï.
V
½
Ù
Û
V
and using (3.8) and (3.9), we obtain
ž

Ï
@ V1> _
Ss0ßØ
V r
Ù
Ï.
A
” Ú
^
^
V1_
]
_
q
Ú
r
Ss:
Ï
^
@ V1_ >
V
<
Ú
V
Ûн
A
Ï.
A
Ù
A
” Ú
^
^
V
_
q
F
Ú
r
Ss
½
½<
½
The result in (3.5) can½< be obtained in a straightforward way
from
the previous expression.
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60
M. ISABEL BUENO, FRANCISCO MARCELLÁN, AND JORGE SÁNCHEZ-RUIZ
3.2. Semiclassical functional z of odd class.
P ROPOSITION 3.5. Let the class Ÿ ž of the functional z be an odd number. The following
explicit algebraic relations between the sequences jp
and jlq
are obtained:
U m
U m
(1) If is an even number and ×\0
Ý­ , then
A
@ V1> _Hà
Ù
Û
Ú
V
^
A V
.
à
” Ú
^
V
V
A
”
A
_
^
V
@ _Hà >
V1
_
Û
Ú
V
:
_
Ù
Û
V
à
^
:
V1_
Ï.
Ú
p
V
Ù
Û
.
Ï.
V1_
”
^½
^
r
.q
V
Ÿ
V
A
.
V
Ï.
V1_Hà
Ï
_
Û
A
” Ú
^
^
q
_
<1Û
V1A _
à
Ÿ ½<
^
V1_
Ù
Ss
r
A
^
V1_
^
Ú
sIq
_
, then
”
^
Û
Ss
r
^
Ss
r
V;_)à
A
F
½
.
à
Ù
:
_
^
V
AV
]
” Ú
^ ^
^
½
½
¬
¯
Os/Ý­
r
q
_
” Ú
^
V
Ss0
r
Ù
^
q
A
Ï
Ù
½
^
V;_
.
à
½ > ½
@ V;_)
à
_
Û
Ú]
A
” Ú
^
V1_
<1
A Û
:
A
½<
½
×\0
A
^
]
.
V
½
^
A
V1_
A
”
V1_
” Ú
^
is an odd number, and
Ï
Ù
Ss0
r
^
r
.
à
Ù½ ^
^
:
(2) If
Ù
Ût½
]Ú
p
A
> à
@ ½ _H
V1
:
Ú
Ú
sIq
_
_
Ss
r
Û
Ss
r
Sst(
r
†¬¯Os/Ý­ and Ê 0
¬ÃOls/Ý­ .
where †
Ê 0
r
r
Proof. Since Ÿ ž is an odd number,  is an odd polynomial [3, Prop. 2.6]. Therefore, there
F
exists another polynomial Ê such that
½
½
½

^
(3.12)
6r
Ÿ
Ÿ
Ê S
 r
G
Ss0\S
s
Ÿ
Ÿ
Since † and ž are odd numbers, so it is . In the sequel, we write †º0ß­ †;
0ß­ Ê :\O .
Ê :‚O and
(1) Assume that is even. Then, we write
0P­-× , for some nonnegative integer × .
From (3.4), we get
F

^
Ss¿×
r
<
@ V1> _Hà
r
V
Ss0
Ù
Û
Ú
V
” Ú
^
^
.
à
Ú
^
k
V
Ss
r
½
Taking into account (1.2), (3.12), and (3.3), we get
F
@ V;> _)à
(3.13)
à
S
Ê

Ss/Ø
r
<
V¨r
Ss0
Ù
Û
Ú
V
” Ú
^
^
à
p
V
.
Ú
Ss
r
½
Consider Proposition 3.2 with YÞ0—­-[Í:\O to obtain,
in a similar way,
A
@ V1> _Hà
(3.14)
à
S
Ê

r
<
SsIØ
V
Ss0
r
Ù
Û
Ú
V
à
A
” Ú
^
^
V1_
.
F
Ú
q
_
r
Ss
½
By multiplying both sides of (3.10) by Sà Ê r Ss , (3.13) and (3.14) lead us to the following
explicit algebraic relation between jlp
and< jlq
,
Um
@ V1> _Hà
Ù
Û
Ú
(3.15)
V
V
à
.
Ù
Û
Ú
V
à
½
p
Ú
r
Ss
A
A
½
@ V;> _)à
0
” Ú
^
^
.
V1_
]
A
” Ú
^
^
Um
_
q
Ú
r
SsB:
A
^
> à
@ V;_)
V
A
Ûe½
Ú
V
à
½
Ù
^
V
.
_
½
½
A
” Ú
^
q
Ú
r
Sst(
ETNA
Kent State University
etna@mcs.kent.edu
CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF ORDER
61
(II)
and the result follows in a straightforward way.
0P­-×á:ßO for some nonnegative integer × . In this
(2) Assume that is odd. Then
case, the term on the left-hand side of (3.7) is an odd polynomial, and taking into account
(1.2), (3.12), and (3.3), we get
(3.16)
à
S
Ê

SsIØ
r
<
Ss0
V r
_
Û
Ú
V
A
Ù
Û
^
Ï.
½
½
F
” Ú
^
V
.
à
A
Ï
> à
V;_)
@
Ú
q
_
Ss
r
½
A similar procedure gives, using Proposition 3.2 with A YZ0ß­w[Ñ:`O ,
A
@ _)à >
V;
(3.17)
à_
S
Ê

SsIØ
r
<
V
Ss0
r
Û
Ú
SØ
à
^
Ú
p
V1_
Ss
r
½
A
^
Ss6:
r
F
” Ú
^
Ï.
]
A
V1_
Ù
.
½
Ss0ßØ
r
V
A
_
Û
V
From (3.2) with YÞ0`­-[Ñ:`O , and (3.3),
(3.18)
Ï
_
F
Ø
V1_
Ss
r
V
A
Replacing (3.16) and (3.17)
into (3.18), the following relation is obtained,
@ _Hà >
V1
Û
Ú
(3.19)
V
0
A
Ï
_
_
Ù
.
Ï.
à
½
@ _)à >
V;
½
Û
Ú
V
_
Ù
Û
r
Ss
]
^
Ï.
½
” Ú
^ ^
V1_
.
à
Ú
p
A
Ï
_
” Ú
^
^
A 1
V _
Û
Ú
q
_
A
^
Ss):
r
@
> à
V1_H
Û
Ú
V1_
V
Ù
” Ú
^
^
V
Ï.
½
and the result follows straightforwardly.
A
Û
.
à
½
A
Ï
_
½
Ú
q
_
SsÐ(
r
½
4. Recurrence relations. In this section we deduce recurrence relations with a finite
number of terms for the sequences jp
, jq
, and jwk
. The number of terms in these
U m
U m
U m
relations depends on the class of the functional z and the degree of the polynomial .

4.1. Recurrence relations for jp
.
Um
P ROPOSITION 4.1. If the class Ÿ ž of z is an even number, then the sequence
satisfies the following [ Ÿ Ê : †E Ê sH:Ãâ ]-term recurrence relation,
r
@

ã ä€å)æ8änç ä€å)æ8èé¿æ8äIê å)æ8è
}ëé æ±ì

D
ã8ä˜åcç ä€å)æ èé
ãä€å)æ8änç ä€å)æ èé
D å)æ8è
/
ëé î ì

½
½ºí
ï
ã8ä˜åGç ä ð
ã8ä€å)æäIç ä ð
_ ð˜ñ åGî8ècò ë æ±ì¿ó

½
½í
ã8ä˜åcç ä€åGî8è ò
ä€å)æ ìIãä€åGç ä€åGî8è ò
_ ó
ä€å
ä˜åGî ì ã ä˜åGîäIç ä˜åcî8è ò îä ê åGî8ècò ë î
½í
½í
E
ä˜å)æ±ìnã8ä€åGç ä˜å)æ èé
ä€å)æ±ìnã8ä˜åGç ä ð
ì
½ºD í

E
ä å
˜
” D
ä€å
ãä€åGç ä˜åBæ8èé
½í
ä˜å
½ºí
ã8ä€åGç ä˜åGî8è
D
ò
ã8ä€åGç ä ð
ê åBæ8ècëé
í
ä€åGî ìnã8ä˜åcîänç ä€åGî8è
_
Um
E

D
ä˜åGî ì/ã8ä˜åGîäIEç äô ð
_
jlp
ò
Eô
E
ê ð

E
D
ê åGî8èGò ë

D
í
í
Ÿ
Ÿ
where †º
Ý-­ .
Ê 0`†Ý-­ and Ê 0
Ÿ
Proof. Assume that ž is an even number. Multiplying both sides of (3.18) by ž

plugging (3.8) and (3.9) into it, we get
S`õö
<
Ú
(4.1)
A
Ï
@ V1> _
Ù
Û
Ï.
V
0
@ V;_ >
½< Ï
<
Ú
V
½<
V1_
Ú
q
_
r
Ss€÷ø
]
A
_
Û
Ï.
” Ú
^
^
A
A
Ù
” Ú
^ ^
^
_
V1_
p
Ú
r
SsÞ:
A
^
V1_
<
Ú
F
Ï
@ V1> _
Û
Ï.
V
½<
Ù
^
” Ú
^
V
p
Ú
r
Ss
<
r
Ss
, and
ETNA
Kent State University
etna@mcs.kent.edu
62
M. ISABEL BUENO, FRANCISCO MARCELLÁN, AND JORGE SÁNCHEZ-RUIZ
Now, we multiply both sides of (3.11) by S and replace (4.1) in it to obtain
Ï
@ V1> _
õö
S
<
Ú
Ù
Û
Ï.
V
½< Ï
@ V1_ >
0
<
Ï.
^
õö
V
” Ú
^ ^
Ù
^
Ú
p
V1_
V
:
Ss ÷ø
r
]
A
_
½<
Ú
p
V
Û
Ú
]
” Ú
^
^
A
A
SsÞ:
r
]
Ï
@ V1> _
Ù
<
Ú
Û
” Ú
^
^
p
V
Ï.
Ú
^
<
r
Ss
F
A
Ù
” Ú
^ ^
^
Ûe½
V
Ú
p
V
Ï.
½<
Ú
p
V
½< Ï
Ú
V
” Ú
^
^
A
Ï.
@ V1_ >
½
V
Ù
Û
V
A
SsB:
r
<
Ú
V;_
_
Ï
@ V1> _
^
Ss ÷ø
r
½
½<
½
From the previous expression we get the ( Ÿ Ê :
†¾
Ê :Ãâ )-term recurrence relation for jlp U m
given in the statement of the proposition.
†º¬\Ols/Ý­ and
P ROPOSITION 4.2. Let the class Ÿ ž of z be an odd number, and put †
Ê 0
r
Ÿ
Ÿ
Ÿ
¬ùOs+Ý-­ . Then the sequence jp
ˆ
:
†º
:‡Os
:ßâ-‹ -term
satisfies
the
following
Ê 0
Ê
Ê
r
r
Um
recurrence relations:
Ý-­ , then
(1) If is an even number,
and ×\0
A
Ù
^
”
^
V1_
Ù ^
0‡ˆ
^
^
V
Ût½
]
Ú
]
:—ˆ
¬
r
V
à
]
SƬ ½
^
ˆ
.
_
V1_
^
V
¬
V1A _
”V
^
Ù
s
”
V
.
V
½<
p ½<
V
” Ú
^
^
¬
Ù
.
<;Û
” Ú
^ ^
^
]
^F
Ù
A
Ù
^
V
^
^
^
V
^
./‹p
V
½
” Ú
^
V
½
”
V
½
Ss
Ss
] r
^
¬
^
V
r
] _Hà
V1
Û
V;A _
V
½
‹p
V;_
A^
à
½
”
^
]
V
.™¬
^
^
V1_
^
^
V
½
Ù
s
V
Ù
sc¬
V
^
^
^
V
^
]
¬
V1_
^
½
(2) If
]
¬
Ù
^ Û
]
]
Ss
r
_
^
SƬ
rA
A
^
^
V
SĬ
r
A
V;_)à
A <1Û
> à
@ V1_H
:
]
p
V1
_
” _ A
^
<1Û
V1_
‹p
Ú
r
Ss
½
V
à
½<
.
Ss
r
½
½
is an odd number, and ×\0
¬¯Os/Ý­
r
@
, then
E
E
ã ä˜å)æ8úIç ä€å)æ èé¿æú êå)æ ûwé¿æn
ëé æ8ä


ä€å)æ8ä 㠘
ä åBæ ì¿ç ä˜E å)æ èé¿æ ì
ä€å)æ±ì 㠘
ä å)æ±ì¿ç ä€å)æ èé¿æ ô ì ê å)
ã ä€å)æ±ì}ç ä€å)æ8èé¿æ±ì
ã ä€å)æúIç ä˜åBæ8èé¿æ±ì
û é¿æn
ëé æ ì
E æ w
D åBæ8ûé}æëé
D


½
½í
½í
ï
ä€å)æä
ä å)æ ì ã ä˜å)æ±ì¿ç ä ð
€
ä˜åBæ ì ˜
ä å ã ä˜åcî ì}ç ä ð ê ð
ã ä€å)æ ì¿ç ä ð
ã ä˜å)æ8úIç ä ð
E
E
_ ð˜ñ åGî8û ò îò ë æ ì¿ó
_
D
D


½
½
í E}ü
í
½ºí
í
ã8ä˜åBæ ì¿ç ä˜åGî8è ò æ±ì
ä€å)æ8ä/ã8ä˜å)æ±ì¿ç ä€åGî8è ò æ ì
ä€å)±
æ ìnã8ä˜å)æ±ì¿ç ä€åGî8è ò æ ì
ä€å)±
æ ì ä€å
ãä€åGî ì¿ç ä€G
å î8è ò æ ì ê åc8
î û ò v
î òë

D
D
ä€å)æ±ì ä€å
ãä€åGî ì¿ç ä˜åGî8è ò î ì¿ê åGî8û ò îò ë î ì
½
½í
½í
½í
í
D
½í
í
Proof. Again, we must distinguish between being odd or even.
(1) Assume that is even. Multiplying both sides of (3.18) by Sà
(3.13) and (3.14) into it, we get
A
@ V1> _Hà
S`õö
(4.2)
0
Ú
Ù
Û
Ú
V
à
½
> à
@ V;_)
_
Û
.
V
à
½
V1_
.
Ú
q
_
r
Ù
^
^
_
V;_
r
<
Ss
and applying
Ss}÷ø
]
A
Ê
A
” Ú
^
^
A

” Ú
^
p
Ú
r
SsB:
A
F
@ V1> _Hà
^
V1_
Ù
Û
Ú
V
à
½
.
” Ú
^
^
V
p
Ú
r
Ss
Now, multiplying both sides of (3.15) by S and using (4.2) in the resulting equation, we obtain
ETNA
Kent State University
etna@mcs.kent.edu
CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF ORDER
the following ×
Ÿ
†1:
r
sB:µâ“0
@ V1> _Hà
Ù
Û
S`õö
Ú
V
.
à
½
> à
@ V;_)
0
” Ú
^
^
A
Ù
Û
V
]
õö
V
Ù
Û
Ú
p
@ V;> _)à
^
SsÞ:
r
V
” Ú
^
^
à
Ú
p
V
.
,
Um
A
^
r
A
Ù
½
V
Ss
r
^
” Ú
^
^
Ú
p
V
.
à
½
Ú
p
V
Ût½
Ú
V
” Ú
^
^
A
.
V
à
½
@ V1_H
> à
A
Ss):
Ù
Û
Ú
V1_
]
@ V1> _Hà
^
Ú
_
½
:
” Ú
^ ^
^
-term recurrence relation for jlp
s
Ss€ø÷
r
V1_
.
à
:Ãâa:
Ê
]
A
_
Ú
Ú
p
V
Ÿ
†a
Ê :
r
63
(II)
Ss ÷ø
r
(
½
½
½
from which the statement (1) of the proposition follows straightforwardly.
(2) Assume now that is odd. Considering (3.10), (3.16), and (3.17), we get
> à
V1_H
@
Û
S\õö
Ú
(4.3)
V
0
A
.
à
½
@ _Hà >
V1
A
Û
à
^
Ï.
Ú
q
_
Ss€÷ø
r
]
A
½
Ù
.
” Ú
^
^
Ú
p
V1_
Ï.
½
” Ú
^
V
Û
V
Ù
Û
Ï ½
_
_
Ú
A
Ï
_
^
SsB:
r
> à
V1_H
@
Û
Ú
V
V
A
Ù
Û
.
à
½
A
Ï
_
^
Ú
p
V
Ï.
½
F
” Ú
^
Ss
r
½
½
½
Now plug the previous result in (3.19). We get the following
A
jlp
recurrence relation for
,
Ÿ
†Õ
Ê :
r
:ßâ :
Ê
s
-term
Um
@ _Hà >
V1
S\õö
Û
Ú
_
.
à
½
@ V1_Hà >
à
½
A
_
Ù
Ï.
½
^
Ú
p
A
Û
Ú
_
Ù
Û
V
à
@ _Hà >
V1
^
.
” Ú
^
^
½
Ú
p
V1_
Ï.
Û
r
^
Ù
Ï.
½
> à
V1_H
@
Û
V
½
A
Ù
Û
.
à
Ss
r
” Ú
^
^
p
V
Ï.
½
Ú
p
A
Ï
_
Ú
V
” Ú
^
^
V;_
.
à
½
SsB:
_
Û
V
]
A
Ï
_
Ú
V1_
A
Ï
_
^
Ss6:
r
_
@ _Hà >
V1
õö
V;_
” Ú
^
V1_HA ý
.
^
:
Ss}ø÷
r
]
Û
A
Ú
p
V1_
^
½Ï
Û
V
” Ú
^
^
Ï.
_
Ú
]
Ù
Û
V
0
A
Ï
_
Ú
Ss€÷ø
r
(
½
½
½
which implies the result (2) in the statement of the proposition.
4.2. Recurrence relations for jlq
. In the sequel, and for the sake of brevity, we will
U m
omit the extended expression of the recurrence relations.
P ROPOSITION 4.3. If the class Ÿ ž of z is an even number, the sequence jq
satisfies
U m
the following [ Ÿ Ê : †Ê s):µâ ]-term recurrence relation,
r
õö
S
A
Ï
@ V;> _
<
Ú
Ù
Û
Ï.
V
½< Ï
@ V1_ >
0
<
Ú
V
:
]
^
V1_
” Ú
^
^
A
V1_
Ú
q
_
r
Ss ÷ø
]
A
A
_
Ù
Û
” Ú
^
^
V;_)ý
Ï.
<
Ú
Û
Ï.
V
½<
Ú
Ù
V1_
^
_
<
Ú
q
r
Ss6:
Ù
Û
V
^
Ï.
½<
@ V1_ >
^
A
Ût½
½<
Ù
Ú
Ss
r
^
_
½
½
A
” Ú
^
V
Ï.
V
q
_
A
Ï
<
Ú
A
” Ú
^
A _
V1
V
]
Ú
A
Ï
@ V1> _
V1_
A
” Ú
^
^
^
SsB:
r
A
Ï
@ V1> _
q
_
_
A
½<
õö
A
q
Ú
r
Ss}øh
÷
(
ETNA
Kent State University
etna@mcs.kent.edu
64
M. ISABEL BUENO, FRANCISCO MARCELLÁN, AND JORGE SÁNCHEZ-RUIZ
Ÿ
Ÿ
Ý-­ .
where †
Ê 0`†Ý-­ and Ê 0
Proof. Replacing (3.11) into (4.1), we get the result.
P ROPOSITION 4.4. Let the class Ÿ ž of z be an odd number, and put †
†º¬\Ols/Ý­ and
Ê 0
r
Ÿ
Ÿ
Ÿ
0
ß
¬
l
O
/
s
Ý
­
.
Then
the
sequence
j
q
satisfies
the
following
ˆ
:
þ
†
É
:
Os™:`â-‹ -term
Ê
Ê
r
U m
r Ê
recurrence relations:
(1) If is even, and ×\0
Ý-­ , then
A
@ V1> _Hà
S\õö
Ù
Û
Ú
V
.
½
> à
@ V1_H
0
^
A
à
V1_
Ù
Û
V
]
^
õö
õö
Û
V
.
à
½
@ _Hà >
V1
à
^
:
õö
V
q
Ù
^
_
Û
Ú
V
à
A
Ù
Û
.
Ú
^
V
Ù
Ûe½
à
A
F
” Ú
^
^
Ú
q
V
.
_
Ss ÷ø
r
½
½
^
½
Û
V
Ú
à
A
” Ú
^
^
V A
Ï.
½
½
@ V1_Hà >
Û
V
.
Ù
½
F
” Ú
^ ^
^
V
^
Ï.
½
Ss
r
A
Ût½
à
½
Ú
q
_
Ï
_
Ú
V
Ù
Û
.
½
^
Ss):
r
A
A
Ï
_
Ú
V1_
]
q
_
> à
V1_H
@
½
½
Ss
r
A
A
Ú
V
A
Ss):
r
” Ú
^
Ï.
½
^
A
Ï
à
½
@ V;_)
> à
Ú
q
_
Ss ÷ø
r
q
_
^
.
V
A
” Ú
^
A _
V1
, then
]
” Ú
^
^
V1_
½
> à
V _H
1
Ù
Û
Ú
Ss):
r
A
Ï.
@
Ú
q
_
½
.
½
^
Ï.
Ú
¬ÃOls/Ý­
r
” Ú
^
V
Û
V
Ù
Û
Ï ½
_
_
Û
Ú
A
A
^
V;_
V
A
Ï
_
@ V1> _Hà
]
_
½
> à
V;_)
]
V1_
à
^
A
^
.
A
SsB:
r
” Ú
^
is odd, and ×`0
Ú
0
q
_
Ù
Û
Ú
V
(2) If
S
Ú
A
@ V1> _Hà
V;_
@
]
_
^
:
Ss€÷ø
r
” Ú
^
V1_Hý
.
à
A
½
Ú
q
_
A
A
_
Ú
A
” Ú
^
Ú
q
_
r
Ss}÷ø
½
½
Proof. We must distinguish again between even or odd. In the case even, replace
(3.15) into (4.2), and in the case
odd, replace (3.19) into (4.3), and so the results are
obtained.
4.3. Recurrence relation for jlk
.
U m
P ROPOSITION 4.5. The sequence jlk
A
A
A
recurrence relation,
A
”
Ù
U-_
U-_
<1Û
@ U_ >
:
r
S
<;Û
”Ú
U]
_
”
Ù ½<
U U
.
r
_
Ù
S
r
.
U
U-_
A
<1Ût½
Ú
:
k
_
U
A
< Û
1
Ù
. s/k
U
½<
U-_
U
½
”Ú
”
Ù
¬
U
.
U
½<
U-_
Ss
]r
r
A
A
”
Ù
Ssc¬
<;Û
U
U
½<
<
Sseˆ ×
r
U-_
r
SsB:
k
×
U
r
U
Ss}‹0
½
A
Ù
<1Û
Ú
.
U
A
”Ú
U
F
.
U
½
½<
A
@ U-> _

A
.
U
½
Proof. Multiplying both sides of (3.4) by S , from (3.2) we get
]
†Õ:ùâ
Ss
r
U-_
<1Û
Ú
sIk
sIk
U-_
½<
A
:
A
½
”
U
] U
¬
A U-_
m
”
Ù
S
r
”Ú
Ù
¬
Ù
¬
Ss0
A
Ÿ
satisfies the following (
U
r
Ss
½
F
Sk
Ú
r
Ss
½<
Applying twice Proposition 3.2 to the previous expression,
we obtain
]
A
@ U-_ >
A
_
Ú
<1Û
.
U
½<
”Ú
Ù
_
U-_
k
A
@ U-_ >
Ú
r
Ss):
U
A
Ú
<1Ûe½
U
.
U
½<
”Ú
Ù
½
½
k
@ U-> _
Ú
r
Ss0
.
U
½<
”Ú
Ù
<;Û
Ú
U
Sk
Ú
r
SsÐ(
)-term
ETNA
Kent State University
etna@mcs.kent.edu
CONTINUOUS SYMMETRIZED SOBOLEV INNER PRODUCTS OF ORDER
(II)
65
from which the result follows straightforwardly.
5. Example: Freud-Sobolev orthogonal polynomials. Consider the following inner
product of type (1.1),
(5.1)
%Ö&)(+*-, .
0
?

2 3
&4*Bÿ
> ?@BABC
58S‡:
<
½
?E
?E

2 3
&HD
* D
ÿ
58SÞ(
½
and let us apply the results obtained in the previous sections to this particular example. The
polynomials jlk
orthogonal with respect to (5.1) are a particular case of the so-called
U m
Freud-Sobolev polynomials [2].

is a semiclassical weight function [2]. In fact, | r Ss
It is well known that | r Ss40‡ÿ
satisfies the Pearson equation (2.3) ½ with 6r Ss0ÇO and  r Ss0 ¬ Sý . Hence, using the
notations in Section 2, †“0—M , Ÿ ž 0—­ , and Ÿ 0—­ . Since Ÿ ž is an even number, we can state that
(1) The sequences jp
and jlq
satisfy ( :“â )-term recurrence relations (see PropoU m
U m
sitions 4.1 and 4.3).
(2) The sequence jlk
satisfies a ( ­
:¯â )-term recurrence relation (see Proposition
U m
4.5).
(3) An explicit algebraic relation between jp
and jq
can be established. This
U m
U m
and :í terms of the sequence jlq
relation involves :—O terms of the sequence jp
U m
U m
(see Proposition 3.4).
Acknowledgements. The authors wish to thank the anonymous referee for his/her careful reading of the manuscript and the comments on it, which were very useful to improve its
presentation. The work of the authors has been partially supported by Dirección General de
Investigación (Ministerio de Ciencia y Tecnología) of Spain under grants BFM 2003-06335C03-02 (M.I.B., F.M., J.S.R.) and BFM2001-3878-C02-01 (J.S.R.), INTAS Research Network NeCCA INTAS 03-51-6637 (F.M.), and the Junta de Andalucía research group FQM0207 (J.S.R).
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