Jacobi-Sobolev-type orthogonal polynomials: second-order differential equation and zeros
Abstract
We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Qn} associated with the inner product 〈p,q〉=∫−11 p(x)q(x)p(x)dx + A1p(1)q(1) + B1p(−1)q(−1) + A2p′(1)q′(1) + B2p′(−1)q′(−1), where p(x) = (1 − x)α(1 + x)β is the Jacobi weight function, α,β> − 1, A1,B1,A2,B2⩾0 and p, q ∈ P, the linear space of polynomials with real coefficients. The hypergeometric representation (6F5) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Qn(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.
Full text
JACOBI-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS:
SECOND ORDER DIFFERENTIAL EQUATION AND ZEROS.
J. A es u, R.
Al a ez-No da se, F. Ma cellan, and K. Pan
P ep in MA/UC3M/7/1997
Dedica ed o P o esso Ma io Rosa io Occo sio on his 65- h bi hday.
Key wo ds and ph ases: O hogonal p olynomials, Jacobi p olynomials,
hyp e geome ic unc ion, Sob ole - yp e o hogonal p olynomials,
WKB me ho d.
AMS (MOS) sub jec classica ion:
33C45, 33A65, 42C05.
Abs ac
We ob ain an explici exp ession o he Sob ole - yp e o hogonal p olynomials
Q
n
(
x
) asso cia ed
wi h he inne p o duc
< p; q >
=
Z
1
?
1
p
(
x
)
q
(
x
)
(
x
)
dx
+
A
1
p
(1)
q
(1) +
B
1
p
(
?
1)
q
(
?
1) +
A
2
p
0
(1)
q
0
(1) +
B
2
p
0
(
?
1)
q
0
(
?
1)
;
whe e
(
x
) = (1
?
x
)
(1 +
x
)
is he Jacobi weigh unc ion,
; >
?
1,
A
1
; B
1
; A
2
; B
2
0 and
p
,
q
2
IP
, he linea space o p olynomials wi h eal co ecien s. The hype geome ic ep esen a ion (
6
F
5
)
and he second o de linea die en ial equa ion ha such polynomials sa is y a e also ob ained.
The asymp o ic b eha iou o such polynomials in [-1, 1] is s udied. Fu he mo e, we ob ain some
es ima es o he la ges ze o o
Q
n
(
x
). Such a ze o is lo ca ed ou side he in e al [-1, 1]. We
deduce his dependence o he masses. Finally, he WKB analysis o he dis ibu ion o ze os is
p esen ed.
1 In o duc ion.
The s udy o some pa icula cases o o hogonal p olynomials in Sob ole spaces has a ac ed
he in e es o se e al au ho s [1], [9], [15], [20], [21] and [25 ]. Pa icula emphasis was gi en o he
so-called classical Sob ole p olynomials o disc e e yp e, i.e., p olynomials o hogonal wi h esp ec o
an inne p o duc
< p; q >
=
Z
p
(
x
)
q
(
x
)
d
(
x
) +
N
X
k
=0
Z
p
(
k
)
(
x
)
q
(
k
)
(
x
)
d
k
(
x
)
;
whe e
d
(
x
) is a classical measu e (Jacobi [1], Gegenbaue [8 ], Lague e [16], Bessel [21 ]) and
d
k
(
x
)
a e Di ac measu es.
Decemb e 14, 1997
1
p p p y g p p
< p; q >
=
Z
1
?
1
p
(
x
)
q
(
x
)
(
x
)
dx
+
A
1
p
(1)
q
(1) +
B
1
p
(
?
1)
q
(
?
1) +
A
2
p
0
(1)
q
0
(1) +
B
2
p
0
(
?
1)
q
0
(
?
1)
;
whe e
(
x
) = (1
?
x
)
(1 +
x
)
is he Jacobi weigh unc ion,
; >
?
1,
A
1
; B
1
; A
2
; B
2
0 and
p
,
q
2
IP
, he linea space o p olynomials wi h eal co ecien s. Some es ima es conce ning o his kind o
p olynomials ha e b een ob ained in [3]. Howe e , he explici o m o hese p olynomials in he gene al
case emains as an op en ques ion as well as he s udy o hei ze os. We a e ying in his pap e o
co e his lack. Mo eo e , some o he usual p op e ies o classical o hogonal p olynomials { sym-
me y p op e y, hei ep esen a ion as hyp e geome ic se ies and he second o de linea die en ial
equa ion { a e ansla ed o he con ex o Sob ole - yp e o ogonali y.
The s uc u e o he pap e is he ollowing. In Sec ion 2 we gi e some esul s conce ning o classical
Jacobi p olynomials. Using hese esul s, in Sec ion 3 we ob ain an explici o mula o he Jacobi-
Sob ole - yp e o hogonal p olynomials in e ms o he classical ones and hei s and second de i a i es
which allows us o deduce a symme y p op e y. In Sec ion 4 we es ablish he ecu ence ela ion
ha he Jacobi-Sob ole - yp e o hogonal p olynomials sa is y, when he masses
A
2
and
B
2
a e b o h
die en om ze o. In Sec ion 5 a ep esen a ion o ou p olynomials as a
6
F
5
hyp e geome ic unc ion
is deduced. Finally, in Sec ion 6 a gene al algo i hm in o de o gene a e he second o de linea
die en ial equa ions ha such p olynomials sa is y is gi en. This esul is basic o he de elopmen
o he Sec ion 8, mo e p ecisely o he WKB me ho d, in o de o ob ain he dis ibu ion o hei
ze os. In Sec ion 7, some asymp o ic o mulas, use ul in he s udy o he ze os, a e p esen ed. Finally,
in Sec ion 8 we ob ain he sp eed o con e gence o hose ze os lo ca ed ou side [-1, 1]. On he o he
hand, we show some g aphics conce ning he WKB densi y as well as he analy ic b eha iou o he
dis ibu ion o ze os o Jacobi-Sob ole - yp e o hogonal p olynomials.
2 Classical Jacobi p olynomials.
In his sec ion we ha e enclosed some o mulas o he classical Jacobi p olynomials which will b e
use ul o ob ain some p op e ies o he Sob ole - yp e o hogonal p olynomials. All he o mulas as
well as some sp ecial p op e ies o he classical Jacobi p olynomials can b e ound in he li e a u e
[23, Chap e 1-2], [27]. In his wo k we will use monic p olynomials, i.e., p olynomials wi h leading
co ecien equal o 1.
The classical Jacobi p olynomials
P
;
n
(
x
) sa is y he o hogonali y ela ion
Z
1
?
1
P
;
n
(
x
)
P
;
m
(
x
)(1
?
x
)
(1 +
x
)
dx
=
nm
d
2
n
;
(1)
whe e
d
2
n
=
jj
P
;
n
(
x
)
jj
2
=
2
2
n
+
+
+1
n
!?(
n
+
+ 1)?(
n
+
+ 1)?(
n
+
+
+ 1)
?(2
n
+
+
+ 1)?(2
n
+
+
+ 2)
:
They a e he p olynomial solu ion o he second o de linea die en ial equa ion o hyp e geome ic
yp e
(
x
)
y
00
(
x
) +
(
x
)
y
0
(
x
) +
n
y
(
x
) = 0
;
(2)
whe e
(
x
) = (1
?
x
2
)
;
(
x
) =
?
?
(
+
+ 2)
x;
n
=
n
(
n
+
+
+ 1)
;
esp ec i ely. No ice ha deg
=2 and deg
=1. Also hey e i y he symme y p op e y
P
;
n
(
x
) = (
?
1)
n
P
;
n
(
?
x
)
;
(3)
2
d
d x
P
;
n
(
x
)
(
P
;
n
(
x
))
(
)
=
n
!
(
n
?
)!
P
+
;
+
n
?
(
x
)
;
wi h
n
and
n
= 0
;
1
;
2
; :::;
(4)
as well as
he h ee- e m ecu ence ela ion
xP
n
(
x
) =
P
;
n
+1
(
x
) +
;
n
P
;
n
(
x
) +
;
n
P
;
n
?
1
(
x
)
;
(5)
whe e
;
n
=
2
?
2
(2
n
+
+
)(2
n
+ 2 +
+
)
;
;
n
=
4
n
(
n
+
)(
n
+
)(
n
+
+
)
(2
n
+
+
?
1)(2
n
+
+
)
2
(2
n
+
+
+ 1)
:
(6)
They a e ep esen ed as he hyp e geome ic se ies
P
;
n
(
x
) =
2
n
(
+ 1)
n
(
n
+
+
+ 1)
n
2
F
1
?
n; n
+
+
+ 1
+ 1
1
?
x
2
!
;
(7)
whe e
p
F
q
a
1
; a
2
; :::; a
p
b
1
; b
2
; :::; b
q
x
!
=
1
X
k
=0
(
a
1
)
k
(
a
2
)
k
(
a
p
)
k
(
b
1
)
k
(
b
2
)
k
(
b
q
)
k
x
k
k
!
;
(8)
and (
a
)
k
is he Po chhamme symb ol o shi ed ac o ial (
a
)
0
:= 1, (
a
)
k
:=
a
(
a
+ 1)(
a
+ 2)
(
a
+
k
?
1) =
=
?(
a
+
k
)
?(
a
)
,
k
= 1
;
2
;
3
; :::
. As a consequence o his ep esen a ion we ge
P
;
n
(1) =
2
n
(
+ 1)
n
(
n
+
+
+ 1)
n
; P
;
n
(
?
1) =
(
?
1)
n
2
n
(
+ 1)
n
(
n
+
+
+ 1)
n
:
(9)
The Ch is oel-Da b oux o mula is
n
?
1
X
m
=0
P
;
m
(
x
)
P
;
m
(
y
)
d
2
m
=
1
x
?
y
P
;
n
(
x
)
P
;
n
?
1
(
y
)
?
P
;
n
?
1
(
x
)
P
;
n
(
y
)
d
2
n
?
1
; n
= 1
;
2
;
3
; :::
(10)
Th oughou he wo k we will deno e
K
;
(
p;q
)
n
(
x; y
) =
n
X
m
=0
(
P
;
m
)
(
p
)
(
x
)(
P
;
m
)
(
q
)
(
y
)
d
2
m
=
@
p
+
q
@ x
p
@ y
q
K
;
n
(
x; y
)
;
(11)
he ke nels o he Jacobi p olynomials, as well as hei de i a i es wi h esp ec o
x
and
y
, esp ec i ely.
By using he symme y p op e y (3) and (11) i is s aigh o wa d o p o e ha he ollowing symme y
p op e ies o he Jacobi ke nels
K
;
n
(
x; y
) =
K
;
n
(
?
x;
?
y
)
;
K
;
(0
;
1)
n
(
x; y
) =
?
K
;
(0
;
1)
n
(
?
x;
?
y
)
;
K
;
(1
;
1)
n
(
x; y
) =
K
;
(1
;
1)
n
(
?
x;
?
y
)
;
(12)
hold. In ou wo k we need he explici exp essions o he ke nels
K
;
n
?
1
(
x;
1),
K
;
(0
;
1)
n
?
1
(
x;
1),
K
;
n
?
1
(
x;
?
1)
and
K
;
(0
;
1)
n
?
1
(
x;
?
1), esp ec i ely. To ob ain hese ke nels we can use he Ch is oel-Da b oux o mula,
he s uc u e ela ion, he h ee- e m ecu ence ela ion and he die en ia ion o mula o classical
monic Jacobi p olynomials, esp ec i ely. The de ailed compu a ion can b e ound in [5]. We will p o ide
3
g p p
no a ion
;
n
= (2
n
+
+
+ 1)
;
n
K
;
n
?
1
(
x;
1) =
P
;
n
(1)
d
2
n
?
1
;
n
h
(1 +
x
)(
P
;
n
)
0
(
x
)
?
nP
;
n
(
x
)
i
;
(13)
K
;
(0
;
1)
n
?
1
(
x;
1) =
(
P
;
n
)
0
(1)
d
2
n
?
1
;
n
h
(1 +
x
)(
P
;
n
)
0
(
x
)
?
nP
;
n
(
x
)
i
?
?
P
;
n
(1)
d
2
n
?
1
;
n
(
+ 1)
h
(1 +
)(
P
;
n
)
0
(
x
) + (
x
+ 1)(
P
;
n
)
00
(
x
)
i
:
(14)
F om he wo p e ious o mulas and using he symme y p op e ies (12), we nd
K
;
n
?
1
(
x;
?
1) =
?
P
;
n
(
?
1)
h
(1
?
x
)(
P
;
n
)
0
(
x
) +
nP
;
n
(
x
)
i
d
2
n
?
1
;
n
;
K
;
(0
;
1)
n
?
1
(
x;
?
1) =
?
(
P
;
n
)
0
(
?
1)
h
(1
?
x
)(
P
;
n
)
0
(
x
) +
nP
;
n
(
x
)
i
d
2
n
?
1
;
n
+
+
P
;
n
(
?
1)
h
(1
?
x
)(
P
;
n
)
00
(
x
)
?
(
+ 1)(
P
;
n
)
0
(
x
)
i
d
2
n
?
1
;
n
(
+ 1)
:
(15)
Also he ollowing alues a e needed [5 ]
K
;
n
?
1
(1
;
1) =
(
P
;
n
(1))
2
n
(
n
+
)
d
2
n
?
1
;
n
(
+ 1)
; K
;
(0
;
1)
n
?
1
(1
;
1) =
(
P
;
n
)
0
(1)
P
;
n
(1)(
n
+
)
d
2
n
?
1
;
n
(
+ 2)(
n
?
1)
?
1
;
K
;
n
?
1
(1
;
?
1) =
?
nP
;
n
(
?
1)
P
;
n
(1)
d
2
n
?
1
;
n
; K
;
(0
;
1)
n
?
1
(1
;
?
1) =
(
P
;
n
)
0
(
?
1)
P
;
n
(1)(1
?
n
)
d
2
n
?
1
;
n
;
K
;
(1
;
1)
n
?
1
(1
;
1) =
P
;
n
(1)(
P
;
n
)
0
(1)(
n
+
)
(
+ 2)(
n
2
+
n
+
n
)
?
(
+ 1)(
+
+ 2)
2
d
2
n
?
1
;
n
(
+ 1)(
+ 2)(
+ 3)(
n
?
1)
?
1
;
K
;
(1
;
1)
n
?
1
(1
;
?
1) =
(
P
;
n
)
0
(
?
1)
P
;
n
(1)(1
?
n
)
n
2
+
n
+
n
?
?
?
2
2
d
2
n
?
1
;
n
(
+ 1)
:
(16)
3 Jacobi-Sob ole - yp e o hogonal p olynomials.
Conside he inne p o duc in he linea space o p olynomials wi h eal co ecien s
< p; q >
=
< p; q >
c
+
A
1
p
(1)
q
(1) +
B
1
p
(
?
1)
q
(
?
1) +
A
2
p
0
(1)
q
0
(1) +
B
2
p
0
(
?
1)
q
0
(
?
1)
;
(17)
whe e
< p; q >
c
is he Jacobi inne p o duc
< p; q >
c
=
Z
1
?
1
p
(
x
)
q
(
x
)(1
?
x
)
(1 +
x
)
dx; >
?
1
; >
?
1
;
(18)
4
1 2 1 2
g
We will deno e
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
g
n
he monic o hogonal p olynomial sequence wi h esp ec o
he inne p o duc (17). They will b e called
Jacobi-Sobole - ype o hogonal polynomials
. Le us now
o nd an explici ep esen a ion o he p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) in e ms o he classical ones.
To ob ain his we w i e he Fou ie expansion o he Jacobi-Sob ole - yp e p olynomials in e ms o he
Jacobi p olynomials
~
Q
n
(
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
P
;
n
(
x
) +
n
?
1
X
k
=0
a
n;k
P
;
k
(
x
)
;
(19)
whe e
P
;
n
(
x
) is he classical Jacobi monic p olynomial o deg ee
n
. To nd he co ecien s
a
n;k
we
can use he o hogonali y o he p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) wi h esp ec o
<; >
, i.e.,
< Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
; P
;
k
(
x
)
>
= 0 0
k < n:
(20)
Thus, acco ding o (17) we nd
< Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
; P
;
k
(
x
)
>
=
< Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
; P
;
k
(
x
)
>
c
+
+
A
1
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1)
P
;
k
(1) +
B
1
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1)
P
k
(
?
1)+
+
A
2
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1)(
P
;
k
)
0
(1) +
B
2
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1)(
P
;
k
)
0
(
?
1)
;
(21)
I we use he decomp osi ion (19) and aking in o accoun (20) we nd he ollowing exp ession o he
co ecien s
a
n;k
a
n;k
=
?
A
1
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1)
P
;
k
(1) +
B
1
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1)
P
;
k
(
?
1)
d
2
k
?
k < n
?
A
2
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1)(
P
;
k
)
0
(1) +
B
2
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1)(
P
;
k
)
0
(
?
1)
d
2
k
;
(22)
whe e
d
2
k
deno es he squa e no m o he classical Jacobi p olynomials (1). Finally, he equa ion (19)
b ecomes
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
P
;
n
(
x
)
?
A
1
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1)
K
;
n
?
1
(
x;
1)
?
?
B
1
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1)
K
;
n
?
1
(
x;
?
1)
?
A
2
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1)
K
;
(0
;
1)
n
?
1
(
x;
1)
?
?
B
2
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1)
K
;
(0
;
1)
n
?
1
(
x;
?
1)
:
(23)
In o de o nd he unknowns
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1),
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1), (
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1) and
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1) we can ake de i a i es in (23) and e alua e he esul ing equa ion, as well
as (23), a
x
= 1 and
x
=
?
1. This leads o a linea sys em o equa ions
IK
~
Q
n
=
Q
n
;
(24)
5
y y
1 2 3 4
k
1
=
0
B
B
B
B
@
1 +
A
1
K
;
n
?
1
(1
;
1)
A
1
K
;
n
?
1
(1
;
?
1)
A
1
K
;
(0
;
1)
n
?
1
(1
;
1)
A
1
K
;
(0
;
1)
n
?
1
(1
;
?
1)
1
C
C
C
C
A
; k
2
=
0
B
B
B
B
@
B
1
K
;
n
?
1
(1
;
?
1)
1 +
B
1
K
;
n
?
1
(
?
1
;
?
1)
B
1
K
;
(0
;
1)
n
?
1
(
?
1
;
1)
B
1
K
;
(0
;
1)
n
?
1
(
?
1
;
?
1)
1
C
C
C
C
A
;
k
3
=
0
B
B
B
B
@
A
2
K
;
(0
;
1)
n
?
1
(1
;
1)
A
2
K
;
(0
;
1)
n
?
1
(
?
1
;
1)
1 +
A
2
K
;
(1
;
1)
n
?
1
(1
;
1)
A
2
K
;
(1
;
1)
n
?
1
(1
;
?
1)
1
C
C
C
C
A
; k
4
=
0
B
B
B
B
@
B
2
K
;
(0
;
1)
n
?
1
(1
;
?
1)
B
2
K
;
(0
;
1)
n
?
1
(
?
1
;
?
1)
B
2
K
;
(1
;
1)
n
?
1
(1
;
?
1)
1 +
B
2
K
;
(1
;
1)
n
?
1
(
?
1
;
?
1)
1
C
C
C
C
A
;
and
~
Q
n
and
Q
n
a e he column ec o s
~
Q
n
=
0
B
B
B
@
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1)
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1)
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1)
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1)
1
C
C
C
A
;
Q
n
=
0
B
B
B
@
P
;
n
(1)
P
;
n
(
?
1)
(
P
;
n
)
0
(1)
(
P
;
n
)
0
(
?
1)
1
C
C
C
A
;
esp ec i ely. Le us deno e
IK
j
(
Q
n
) he ma ix ob ained subs i u ing he
j
column in
IK
by
Q
n
. Then,
om he C ame 's, ule he sys em (24) has a unique solu ion i and only i he de e minan o
IK
do es
no anish. Mo eo e , he solu ion is gi en by
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1) =
de
IK
1
(
Q
n
)
de
IK
; Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1) =
de
IK
2
(
Q
n
)
de
IK
;
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1) =
de
IK
3
(
Q
n
)
de
IK
;
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1) =
de
IK
4
(
Q
n
)
de
IK
:
(25)
He e we wan o ema k ha , since ou p olynomials a e o hogonal wi h esp ec o (17), hen he
p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) exis o all alues o he nonnega i e masses
A
1
,
B
1
,
A
2
and
B
2
. In
pa icula his implies ha de
IK
6
= 0. This si ua ion is e y die en om one s udied in [5] whe e
he p olynomials a e o hogonal wi h esp ec o a linea unc ional which is no p osi i e deni e (in
gene al i is no a quasi-deni e linea uc ional).
P op osi ion 1
The ol lowing symme y p ope y o he Jacobi-Sobole polynomials holds
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
x
) = (
?
1)
n
Q
;;B
1
;A
1
;B
2
;A
2
n
(
x
)
:
(26)
P o o
: Le us deno e he de e minan o
IK
by
4
; ;A
1
;B
1
;A
2
;B
2
n
and he de e minan o
IK
j
(
Q
n
) by
4
; ;A
1
;B
1
;A
2
;B
2
n;j
(
Q
n
). I we in e change in
IK
2
(
Q
n
) he s and second columns and he s and
second ows, he hi d and ou h columns and he hi d and ou h ows, esp ec i ely, and hen we
use he symme y p op e y o Jacobi p olynomials (9) and hei ke nels (12) we nd he ollowing
ela ion o he de e minan s
4
;;B
1
;A
1
;B
2
;A
2
n;
2
(
Q
n
) = (
?
1)
n
4
; ;A
1
;B
1
;A
2
;B
2
n;
1
(
Q
n
)
:
I we handle wi h he same ows and columns bu in
IK
we ge
4
; ;A
1
;B
1
;A
2
;B
2
n
=
4
;;B
1
;A
1
;B
2
;A
2
n
:
Then, om (25) we ob ain
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
?
1) = (
?
1)
n
Q
;;B
1
;A
1
;B
2
;A
2
n
(1)
:
(27)
6
y
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(
?
1) = (
?
1)
n
?
1
(
Q
;;B
1
;A
1
;B
2
;A
2
n
)
0
(1)
:
(28)
Now, i we p o ide he change o pa ame e s
$
,
A
1
$
B
1
and
A
2
$
B
2
in (23) and hen use he
symme y p op e ies o he Jacobi ke nels (12) and (27)-(28) he p op osi ion holds.
Le us now o ob ain an explici o mula o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) in e ms o he classical Jacobi
p olynomials and hei s and second de i a i es. We s a om o mula (23) whe e we subs i u e
he ke nels by hei explici exp essions (13)-(15) and use he o mulas (25). This leads o he ollowing.
P op osi ion 2
The Jacobi-Sobole o hogonal polynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
can be gi en in e ms
o he classical Jacobi polynomials and hei s and second de i a i es
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) = (1 +
n
n
+
n
n
)
P
;
n
(
x
) + [
n
(1
?
x
)
?
n
(1 +
x
)+
+(
+ 1)
n
+ (
+ 1)
!
n
](
P
;
n
(
x
))
0
+ [
n
(1 +
x
)
?
!
n
(1
?
x
)] (
P
;
n
(
x
))
00
;
(29)
whe e
n
=
B
1
C
;;B
1
;A
1
;B
2
;A
2
n
+
B
2
D
;;B
1
;A
1
;B
2
;A
2
n
;
n
=
A
1
C
; ;A
1
;B
1
;A
2
;B
2
n
+
A
2
D
; ;A
1
;B
1
;A
2
;B
2
n
;
(30)
n
=
A
2
E
; ;A
1
;B
1
;A
2
;B
2
n
; !
n
=
B
2
E
;;B
1
;A
1
;B
2
;A
2
n
;
(31)
and
C
; ;A
1
;B
1
;A
2
;B
2
n
=
Q
; ;A
1
;B
1
;A
2
;B
2
n
(1)
P
;
n
(1)
d
2
n
?
1
;
n
;
D
; ;A
1
;B
1
;A
2
;B
2
n
=
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1)(
P
;
n
)
0
(1)
d
2
n
?
1
;
n
;
E
; ;A
1
;B
1
;A
2
;B
2
n
=
(
Q
; ;A
1
;B
1
;A
2
;B
2
n
)
0
(1)
P
;
n
(1)
d
2
n
?
1
;
n
(1 +
)
:
(32)
No ice ha he cons an s
n
;
n
;
n
and
!
n
depend on
n; ;
and he masses
A
1
; B
1
; A
2
and
B
2
.
In he nex Sec ion we will es ablish he ecu ence ela ion ha he p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
sa is y. No ice ha , since he ma ix o he momen s o he inne p o duc dened by (17) is no
o Hankel yp e b ecause
< x ; x >
6
=
<
1
; x
2
>
, hen he Sob ole - yp e o hogonal p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) don' sa is y a h ee- e m ecu ence ela ion. In ac hey will sa is y a se en-
e m ecu ence ela ion (see [15 ]).
4 The se en- e m ecu ence ela ion o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
.
He e we will p o e ha he p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) sa is y a se en- e m ecu ence ela ion.
In ac , i 's s aigh o wa d o p o e ha he mul iplica ion op e a o by (
x
2
?
1)
2
is symme ic wi h
esp ec o (17). The p oblem is o nd a p olynomial op e a o o he lowes deg ee which b e symme ic
wi h esp ec o he Sob ole inne p o duc (17).
Cases:
1. I
A
2
=
B
2
= 0 we ha e a s anda d inne p o duc , hence he mul iplica ion op e a o by
x
is
symme ic.
7
2 2
6
p p y ( y
ob ain a e- e m ecu ence ela ion.
3. I
A
2
6
= 0 and
B
2
= 0 hen he mul iplica ion op e a o b
y (
x
?
1)
2
is symme ic. Hence we
ob ain a e- e m ecu ence ela ion.
The e is ano he in e es ing case, when he masses
A
2
and
B
2
a e b o h die en om ze o. This
si ua ion will b e conside ed b elow.
We assume ha
A
2
6
= 0 and
B
2
6
= 0. In pa icula , om (17) we ge
< hp; q >
=
< p; hq > p; q
2
IP
;
(33)
o some p olynomial
h
(
x
) o deg ee less han o equal o ou . This implies ha
A
2
(
hp
)
0
(1)
q
0
(1) +
B
2
(
hp
)
0
(
?
1)
q
0
(
?
1) =
A
2
(
hq
)
0
(1)
p
0
(1) +
B
2
(
hq
)
0
(
?
1)
p
0
(
?
1)
;
8
p; q
2
IP
:
(34)
The e o e
A
2
h
0
(1)
p
(1)
q
0
(1) +
B
2
h
0
(
?
1)
p
(
?
1)
q
0
(
?
1) =
A
2
h
0
(1)
q
(1)
p
0
(1) +
B
2
h
0
(
?
1)
q
(
?
1)
p
0
(
?
1)
;
(35)
o , equi alen ly,
A
2
h
0
(1)
p
(1)
q
0
(1)
?
p
0
(1)
q
(1)
+
B
2
h
0
(
?
1)
p
(
?
1)
q
0
(
?
1)
?
q
(
?
1)
p
0
(
?
1)
= 0
;
8
p; q
2
IP
:
(36)
I
p
(
x
) = 1 and
q
(
x
) =
x
he equa ion (36) yields
A
2
h
0
(1) +
B
2
h
0
(
?
1) = 0
:
(37)
I
p
(
x
) = 1 and
q
(
x
) =
x
2
he equa ion (36) leads
2
A
2
h
0
(1)
?
2
B
2
h
0
(
?
1) = 0
:
(38)
Thus, om (37)-(38) we ge
(
A
2
h
0
(1) +
B
2
h
0
(
?
1) = 0
;
A
2
h
0
(1)
?
B
2
h
0
(
?
1) = 0
:
(39)
As
A
2
6
= 0 and
B
2
6
= 0 =
)
h
0
(1) =
h
0
(
?
1) = 0, hence
h
0
(
x
) = (
x
2
?
1)
(
x
). The minimal choice o
(
x
) is, in his si ua ion,
(
x
)
1. The e o e
h
(
x
) =
x
3
3
?
x
+
a
(40)
o , equi alen ly,
h
(
x
) =
x
3
?
3
x
+
b:
(41)
In o de o op e a e wi h
h
(
x
) we pu
b
= 0. In such a way we can gua an ee ha
h
(
x
) =
x
3
?
3
x
leads o he sea ched symme ic op e a o on
IP
, when
A
2
6
= 0 and
B
2
6
= 0. This ac allows o w i e a
se en- e m ecu ence ela ion o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
). In ac , om
(
x
3
?
3
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
n
+3
X
j
=0
nj
Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
:
(42)
and aking in o accoun ha
nj
=
<
(
x
3
?
3
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
>
< Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
>
=
=
< Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
;
(
x
3
?
3
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
>
< Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
>
= 0
;
i
j < n
?
3
;
(43)
8
(
x
3
?
3
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
n
+3
X
j
=
n
?
3
nj
Q
; ;A
1
;B
1
;A
2
;B
2
j
(
x
)
;
(44)
whe e
n;n
?
3
=
<
(
x
3
?
3
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
>
< Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
>
=
=
< Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
;
(
x
3
?
3
x
)
Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
>
< Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
>
=
=
< Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
>
< Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
; Q
; ;A
1
;B
1
;A
2
;B
2
n
?
3
(
x
)
>
>
0
:
(45)
5 Rep esen a ion as hyp e geome ic se ies.
He e we will p o e he ollowing p op osi ion
P op osi ion 3
The o hogonal polynomial
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
is, up o a cons an ac o , a gene al-
ized hype geome ic se ies. Mo e p ecisely,
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
2
n
?
3
(
+ 3)
n
?
3
4
(0)
(
n
+
+
+ 1)
n
6
F
5
?
n;n
+
+
+1
;
0
+1
;
1
+1
;
2
+1
;
3
+1
+3
;
0
;
1
;
2
;
3
1
?
x
2
!
;
(46)
whe e
4
(0)
is gi en in
(52)
and he coecien s
?
0
,
?
1
,
?
2
and
?
3
a e he ze os o a polynomial
o ou h deg ee a
k
(see o mula
(49)
om below). In gene al, hey a e complex numbe s. I o
some
i
= 0
;
1
;
2
;
3
,
?
i
is a nega i e in ege numbe we need o ake he analy ic con inua ion o he
hype geome ic se ies
(46)
.
The ep esen a ion (46) can b e conside ed as a gene aliza ion o he ep esen a ion as hyp e geome ic
se ies o he Jacobi p olynomials.
P o o :
Using (4)-(5) we can ew i e (29) as ollows
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
A
n
P
;
n
(
x
) +
nB
n
P
+1
;
+1
n
?
1
(
x
) +
nC
n
P
+1
;
+1
n
(
x
) +
nD
n
P
+1
;
+1
n
?
2
(
x
)+
+
n
(
n
?
1)
E
n
P
+2
;
+2
n
?
2
(
x
) +
n
(
n
?
1)
F
n
P
+2
;
+2
n
?
1
(
x
) +
n
(
n
?
1)
G
n
P
+2
;
+2
n
?
3
(
x
)
;
(47)
whe e
A
n
= 1
?
nC
n
; B
n
=
n
?
n
+
C
n
+1
;
+1
n
?
1
; C
n
=
?
(
n
+
n
)
; D
n
=
C
n
+1
;
+1
n
?
1
;
E
n
=
n
?
!
n
+
F
n
+2
;
+2
n
?
2
; F
n
=
n
+
!
n
; G
n
=
F
n
+2
;
+2
n
?
2
:
(48)
Subs i u ing he hyp e geome ic ep esen a ion o he Jacobi p olynomials (7) in (47) we nd
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) =
2
n
?
3
(
+ 3)
n
?
3
(
n
+
+
+ 1)
n
1
X
k
=0
"
8
A
n
(
n
+
)(
k
+
+ 1)(
k
+
+ 2)
?
?
4
B
n
(
n
+
)(
k
?
n
)(
k
+
n
+
+
+ 1)(
k
+
+ 2)+
+
8
nC
n
(
n
+
)(
n
+
+ 1)(
k
+
n
+
+
+ 1)(
k
+
n
+
+
+ 2)(
k
+
+ 2)
(2
n
+
+
+ 1)(2
n
+
+
+ 2)
+
9
pp
In his sec ion we will apply he so-called semiclassical o WKB app oxima ion (see [7], [29] and
e e ences he ein) o nd he WKB densi y o ze os o he p olynomials
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
). Le us
deno e hese ze os by
x
n;i
g
n
i
=1
. Then he co esp onding dis ibu ion unc ion o ze os is gi en by
n
(
x
) =
1
n
n
X
i
=1
(
x
?
x
n;i
)
:
(84)
He e we will use he me ho d p esen ed in [29] in o de o ob ain he WKB densi y o ze os, which
gi es an app oxima e analy ic exp ession o he densi y o ze os o he solu ions o any linea second
o de die en ial equa ion wi h p olynomial co ecien s. In pa icula we will conside (60)
~
(
x
)
y
00
+ ~
(
x
)
y
0
+
~
(
x
)
y
= 0
:
(85)
The key s ep is he ollowing
Theo em 3
([29])
Le
S
(
x
)
and
(
x
)
be he unc ions
S
(
x
) =
1
4 ~
(
x
)
2
h
2 ~
(
x
)
2
~
(
x
)
?
~
0
(
x
)
+ ~
(
x
)
?
2 ~
0
(
x
)
?
~
(
x
)
i
;
(86)
(
x
) =
1
4[
S
(
x
)]
2
(
5[
S
0
(
x
)]
2
4[
S
(
x
)]
?
S
00
(
x
)
)
=
P
(
x; n
)
Q
(
x; n
)
;
(87)
whe e
P
(
x; n
)
and
Q
(
x; n
)
a e polynomials in
x
as wel l as in
n
. I he condi ion
sup
x
2
X
j
(
x
)
j
<<
1
holds,
hen he semiclassical o WKB densi y o ze os o he solu ions o
(85)
is gi en by
W K B
(
x
) =
1
q
S
(
x
)
; x
2
X
IR
;
(88)
in e e y in e al
X
whe e he unc ion
S
(
x
)
is posi i e.
Using he ab o e algo i hm, he compu a ions ha e b een p e o med by using he symb olic com-
pu e algeb a package
Ma hema ica
[28]. Fi s o all we check he condi ions o he Theo em nding
ha in he conside ed case
n
?
1
, so he Theo em can b e applied o
n
la ge enough. The explici
exp ession o
W K B
(
x
) gi en by (88) is ex emely la ge and we will omi i he e. I is s aigh o wa d
o see ha i we ake he limi
A
1
; A
2
; B
1
; B
2
!
0 in he esul ing exp ession o
W K B
(
x
) we eco e
he classical exp ession o he Jacobi p olynomials [29]. We will p o ide he e some g aphics o he
no malized
W K B
(
x
) unc ion. In Figu e 1 he WKB densi y o ze os o he Jacobi-Sob ole - yp e
o hogonal p olynomials app ea s. We ha e used he o mulas (60), (69), (86) and (88) and plo ed he
no malized Densi y unc ion o
n
= 10
4
in ou die en cases wi h se e al alues o he pa ame e s
and
(
=
= 0,
=
=
?
1
2
,
=
= 5 and nonsymme ic case
= 0 and
= 1). In Figu e
2 app ea s he WKB densi y o ze os o he same alues o
and
and
n
= 10
5
. In Figu e 3 we
ep esen he WKB densi y o ze os o
n
= 10
6
and he same alues o he pa ame e s
and
.
Finally, in Figu e 4 is shown
W K B
(
x
), o
n
= 10
7
wi h he ab o e alues o
and
. Clea , in each
Figu e om he b o om o he op, is dis inguishible he case
=
= 0, while he emaining cases
b eha e almos equal. Some nume ical es s based on he compu a ion o he numb e
N
o ze os in
he in e al (
?
1
10
;
1
10
) by using he exp ession
N
R
1
=
10
?
1
=
10
W K B
(
x
)
dx
o b o h amilies o o hogonal
p olynomials
P
;
n
(
x
) and
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) show ha hei
global
sp ec al p op e ies a e he same.
This esul is in acco dance wi h he nex one.
16
n
g
n
ze o o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
. Then
n
?!
1
p
1
?
x
2
;
in he weak s a opology.
P o o :
F om (6) and (69), we ge
A
n
= 1
?
+
+ 4
n
+
o
1
n
; E
n
=
1 + 5
+
2
?
5
?
2
n
4
+
o
1
n
4
;
B
n
=
2(
?
)
n
2
+
2
?
(19 +
(7 +
)) + 2
(5 +
)
?
5
2
?
3
+ 3 (4 +
)
n
4
+
o
1
n
4
C
n
=
2
n
2
(
+
+ 4) +
o
1
n
2
; D
n
=
1
2
n
2
(
+
+ 6) +
o
1
n
2
;
F
n
=
12 +
(5 +
) +
(5 +
)
2
n
4
+
o
1
n
4
; G
n
=
2(12 +
(5 +
) +
(5 +
))
n
4
+
o
1
n
4
:
Using (47),
jj
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
jj
[
?
1
;
1]
A
n
jj
P
;
n
(
x
)
jj
[
?
1
;
1]
+
nB
n
jj
P
+1
;
+1
n
?
1
(
x
)
jj
[
?
1
;
1]
+
nC
n
jj
P
+1
;
+1
n
(
x
)
jj
[
?
1
;
1]
+
nD
n
jj
P
+1
;
+1
n
?
2
(
x
)
jj
[
?
1
;
1]
+
n
(
n
?
1)
E
n
jj
P
+2
;
+2
n
?
2
(
x
)
jj
[
?
1
;
1]
+
n
(
n
?
1)
F
n
jj
P
+2
;
+2
n
?
1
(
x
)
jj
[
?
1
;
1]
+
n
(
n
?
1)
G
n
jj
P
+2
;
+2
n
?
3
(
x
)
jj
[
?
1
;
1]
;
(89)
whe e
jj jj
[
?
1
;
1]
deno es he sup-no m in he in e al [-1, 1].
Because o
jj
P
;
n
(
x
)
jj
1
n
[
?
1
;
1]
1
2
(see [27]), we deduce
lim
n
!1
jj
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
)
jj
1
n
[
?
1
;
1]
1
2
:
(90)
Thus, om Theo em 2.1 in [10]
n
?!
1
p
1
?
x
2
:
(91)
ACKNOWLEDGEMENTS
Pa o his wo k was p o ided du ing he s ay o he second au ho in he Uni e si y o Ams e dam.
He is e y g a e ul o he Depa men o Ma hema ics o he Uni e si y o Ams e dam o his kind
hospi ali y. The esea ch o he s au ho (JA) was supp o ed by a g an o Minis e io de Educacion
y Cul u a (MEC) o Spain. The esea ch o he h ee s au ho s (JA, RAN and FM) was supp o ed
by Di eccion Gene al de Ense ~nanza Sup e io (DGES) o Spain unde g an PB 96-0120-C03-01. The
au ho s a e e y g a e ul o he unknown e e ees o hei help ul ema ks and o help us o co ec
some missp in s and e o s and signican ly imp o e he pap e .
17
[1] M. Al a o, F. Ma cellan, M. Rezola, and A. Ron eaux:
On o hogonal polynomials o Sobole
ype: Algeb aic p ope ies and ze os.
SIAM J. Ma h. Anal.
23
(1992), 737-757.
[2] M. Al a o, F. Ma cellan, M. Rezola, and A. Ron eaux:
Sobole - ype o hogonal polynomials: The
nondiagonal case.
J. o App ox. Theo y.
83
(1995), 266-287.
[3] M. Al a o, F. Ma cellan, and M. Rezola:
Es ima es o Jacobi-Sobole ype o hogonal polynomials
(1996). Publicaciones del Semina io Ma ema ico Ga ca Galdeano. Uni e sidad de Za agoza. Se ie
I I.
[4] R.
Al
a ez-No da se and F. Ma cellan:
On he Modica ions o Classical O hogonal Polynomials:
The Symme ic Case.
App ox. Th. and Appl. (1997). (In p ess)
[5] R.
Al a ez-No da se, J. A es u, and F. Ma cellan :
A Gene aliza ion o he Jacobi-Koo n-
winde Polynomials.
(Submi ed) P ep in Dep . Ma ema icas (Uni . Ca los I I I de Mad id)
MA/UC3M/6/1997 (1997)
[6] R.
Al a ez-No da se and A. Za zo:
On some modica ions o classical o hogonal polynomials:
Die en ial and spec al p ope ies.
P ep in 1997.
[7] E.R. A iola, A. Za zo, and J.S. Dehesa:
Spec al P ope ies o he biconuen Heun die en ial
equa ion.
J. Compu . Appl. Ma h.
37
(1991), 161-169.
[8] H. Ba inck and H. G. Meije :
O hogonal Polynomials wi h Respec o a Symme ic Inne P oduc
In ol ing De i a i es.
Appl. Analysis
33
, (1989), 103-117.
[9] H. Ba inck and H. G. Meije :
On O hogonal Polynomials wi h Respec o an Inne P oduc
In ol ing De i a i es: Ze os and Recu ence Rela ions.
Indag. Ma h. (N.S)
1
, (1990), 7-14.
[10] H. P. Bla , E. B. Sa and M. Simkani:
Jen zsch-Szego ype Theo ems o he ze os o bes
app oximan s.
J. London Ma h. So c.
38
, (1988), 307-316.
[11] T. S. Chiha a:
O hogonal polynomials and measu es wi h end poin masses.
Ro cky Moun . J. o
Ma h.
15
,(1985), 705-719.
[12] T.S. Chiha a:
An In o duc ion o O hogonal Polynomials.
(Go don and B each, New
Yo k, 1978).
[13] N. D adi:
Su l'adjonc ion de deux masses de Di ac a une o me lineai e egulie e quelconque.
These Do c o a de l'Uni e si e Pie e e Ma ie Cu ie. Pa is, 1990.
[14] N. D adi and P. Ma oni:
Su l'adjonc ion de deux masses de Di ac a une o me egulie e quel-
conque.
In
Polinomios o ogonales y sus aplicaciones.
A. Cacha ei o and E. Go doy Eds.
Ac as del V Simp osium. Uni e sidad de San iago. Vigo 1988, 83-90.
[15] W.D. E ans, L.L. Li lejohn, F. Ma cellan, C. Ma ke , and A. Ron eaux:
On ecu ence ela ions
o Sobole o hogonal polynomials.
SIAM J. Ma h. Anal.
26
, (1995), 446-467.
[16] R. Ko eko ek and H.G. Meije :
A gene aliza ion o Lague e polynomials.
SIAM J. Ma h. Anal.
24
, (1993), 768-782.
[17] T. H. Ko o nwinde :
O hogonal polynomials wi h weigh unc ion
(1
?
x
)
(1 +
x
)
+
M
(
x
+ 1) +
N
(
x
?
1)
.
Canad. Ma h. Bull,
27
, (1984), 205-214.
18
[ ] p
y p p y g
wi h espec o a disc e e Sobole inne p oduc .
Cons . App ox.
11
, (1995), 107-137.
[19] F. Ma cellan and P. Ma oni:
Su l'adjonc ion d'une masse de Di ac a une o me egulie e e
semi-classique.
Ann. Ma . Pu a ed Appl.,
IV
, CLXI I, (1992), 1-22.
[20] F. Ma cellan and A. Ron eaux :
On a class o polynomials o hogonal wi h espec o a Sobole
inne p oduc .
Indag. Ma h. (N.S.)
1
, (1990), 451-464.
[21] F. Ma cellan, T.E. Pe ez, and M.A. Pi ~na :
Regula Sobole ype o hogonal polynomials: The
Bessel case.
Ro cky Moun . J. o Ma h.
25
, (1995), 1431-1457.
[22] P. Ne ai:
O hogonal Polynomials.
Memoi s o he Ame . Ma h. So c.
213
, Ame . Ma h. So c.
P o idence, Rho de Island, 1979.
[23] A. F. Niki o o and V. B. U a o :
Sp ecial Func ions o Ma hema ical Physics.
Bi khause
Ve lag, Basel, 1988.
[24] E. M. Nikishin and V. N. So okin:
Ra ional App oxima ions and O hogonali y.
T ans. o
Ma h. Monog aphs,
92
, Ame . Ma h. So c., P o idence, Rho de Island, 1991.
[25] M.A. Pi ~na :
Polinomios o ogonales ipo Sobole . Aplicaciones.
Do c o al Disse a ion, Uni e si y
o G anada, Spain, 1992.
[26] F. W. J. Ol e :
Asymp o ics and Sp ecial Func ions.
Academic P ess Inc., New Yo k, 1974.
[27] G. Szego:
O hogonal Polynomials.
Ame . Ma h. So c. Collo q. Publ.,
23
, Ame . Ma h. So c.,
P o idence, Rho de Island, 1975 (4 h edi ion).
[28] S. Wol am:
MATHEMATICA
. A sys em o doing Ma hema ics by Compu e
. Addison-
Wesley Publishing Co., New Yo k, 1991.
[29] A. Za zo and J.S. Dehesa:
Spec al P ope ies o solu ions o hype geome ic- ype die en ial
equa ions.
J. Compu . Appl. Ma h.
50
(1994), 613-623.
J. A es u
y
E-mail:
[email p o ec ed]
R.
Al a ez-No da se
y
;
E-mail:
[email p o ec ed]
F. Ma cellan
y
E-mail:
[email p o ec ed]
K. Pan
E-mail:
[email p o ec ed]
y
Depa amen o de Ma ema icas. Escuela Poli ecnica Sup e io .
Uni e sidad Ca los I I I de Mad id. Bu a que 15, 28911, Leganes, Mad id.
Ins i u o Ca los I de Fsica Teo ica y Compu acional
Uni e sidad de G anada E-18071, G anada
Depa men o Ma hema ics and Compu e Science,
Ba y Uni e si y, Miami Sho es, Flo ida 33161-6695. USA.
19
-1 -0.5 0.5 1
0.5
1
1.5
2
2.5
Figu e 1:
WKB Densi y o ze os o
n
= 10
4
o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) wi h
x
2
[
?
0
:
99
;
0
:
99]
.
-1 -0.5 0.5 1
0.5
1
1.5
2
2.5
Figu e 2:
WKB Densi y o ze os o
n
= 10
5
o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) wi h
x
2
[
?
0
:
99
;
0
:
99]
.
-1 -0.5 0.5 1
0.5
1
1.5
2
Figu e 3:
WKB Densi y o ze os o
n
= 10
6
o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) wi h
x
2
[
?
0
:
986
;
0
:
986].
-1 -0.5 0.5 1
0.5
1
1.5
2
2.5
Figu e 4:
WKB Densi y o ze os o
n
= 10
7
o
Q
; ;A
1
;B
1
;A
2
;B
2
n
(
x
) wi h
x
2
[
?
0
:
986
;
0
:
986]
.
Figu e 5: Compa ison o he nume ical compu a ion esul s.
20