WKB approximation and Krall-Type orthogonal polynomials
Abstract
We give an uni ed approach to the Krall-type polynomials orthogonal with respect to a positive measure consisting of an absolutely continuous one \perturbed" by the addition of one or more delta Dirac functions. Some examples studied by di erent authors are considered from an unique point of view. Also some properties of the Krall polynomials are studied. The three-term recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order di erential equations that such polynomials satisfy which allows us to obtain the central moments and the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings and tridiagonal periodic matrices are also studied.
Full text
WKB APPROXIMATION AND
KRALL-TYPE ORTHOGONAL
POLYNOMIALS .
R.
Al a ez-No da se
, F. Ma cellan
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.
J. Pe onilho
z
Dp o. de Ma ema ica, Faculdade de Ci^encias e Tecnologia,
Uni e sidade de Coimb a, Apa ado 3008, 3000 Coimb a, Po ugal.
13 Sep emb e 1996
Key wo ds and ph ases: O hogonal p olynomials, K all p olynomials, WKB App oxima ion,
idiagonal ma ices, dis ibu ion o ze os.
AMS (MOS) sub jec classica ion:
33C45, 33A65, 42C05.
Abs ac
We gi e an unied app oach o he K all- yp e p olynomial s o hogonal wi h esp ec
o a p osi i e measu e consis ing o an absolu ely con inuous one
pe u bed"
by he
addi ion o one o mo e del a Di ac unc ions. Some examples s udied by die en
au ho s a e conside ed om an unique p oin o iew. Also some p op e ies o he K all
p olynomial s a e s udied. The h ee- e m ecu ence ela ion is calcula ed explici l y, as
well as some asymp o ic o mulas. Wi h sp ecial emphasis will b e conside ed he second
o de die en ial equa ions ha such p olynomials sa is y which allows us o ob ain
he cen al momen s and he WKB app oxima ion o he dis ibu ion o ze os. Some
examples coming om quad a ic ans o ma ion p olynomial mappings and idiagonal
p e io dic ma ices a e also s udied.
1 In o duc ion.
In his wo k we p esen a su ey and some new esul s ela i e o he K all yp e o hogonal
p olynomials, i.e., p olynomials wi h a e o hogonal wi h esp ec o an absolu ely con inuous
measu e
pe u bed"
by he addi ion o one o mo e del a Di ac unc ions. These p olynomials
we e s ly s udied in 1940 by H. L. K all [29]. Mo e exac ly H. L. K all in his 1940`s wo k
E-mail: ena [email protected]
y
E-mail: pacoma [email protected]
z
E-mail: p e onilho@ma .uc.p
1
has ob ained h ee new classes o p olynomials o hogonal wi h esp ec o measu es which
a e no absolu ely con inuous wi h esp ec o he Leb esgue measu e. In ac , his s udy is
ela ed o an ex ension o he e y well known cha ac e iza ion o classical o hogonal p oly-
nomials by S. Bo chne . This kind o measu es was no conside ed in [39]. Mo eo e , in his
pap e H. L. K all ob ain ha hese h ee new amilies o o hogonal p olynomials sa is y a
ou h o de die en ial equa ion wi h p olynomial co ecien s. The co esp onding measu es
a e gi en in able 1. A die en app oach o his sub jec was p esen ed in [28].
Table 1
: The classical K all p olynomials [28], [29]
P
n
g
measu e
d supp
(
)
Lague e- yp e
e
x
dx
+
M
(
x
)
; M >
0 [0
;
1
)
Legend e- yp e
2
dx
+
(
x
1)
2
+
(
x
+ 1)
2
; >
0[
1
;
1]
Jacobi- yp e (1
x
)
dx
+
M
(
x
)
; M >
0
; >
1 [0
;
1]
The analysis o p op e ies o p olynomials o hogonal wi h esp ec o a p e u ba ion o a
measu e ia he addi ion o mass p oin s was in o duced by P.Ne ai [35]. The e he asymp-
o ic p op e ies o he new p olynomials ha e b een conside ed. In pa icula , he p o ed he
dep endence o such p op e ies in e ms o he lo ca ion o he mass p oin s wi h esp ec o
he supp o o he measu e. Pa icula emphasis was gi en o measu es supp o ed in [
1
;
1]
and sa is ying some ex a condi ions in e ms o he pa ame e s o he h ee- e m ecu ence
ela ion ha he co esp onding sequence o o hogonal p olynomials sa ises.
The analysis o algeb aic p op e ies o such p olynomials a ac ed he in e es o se e al
esea che s. A gene al analysis when a mo dica ion o a linea unc ional in he linea space
o p olynomials wi h eal co ecien s ia he addi ion o one del a Di ac measu e was s a ed
by Chiha a [13] in he p osi i e deni e case and Ma cellan and Ma oni [31] o quasi-deni e
linea unc ionals. F om he p oin o iew o die en ial equa ions see [34]. Fo wo p oin
masses he e exis e y ew examples in he li e a u e (see [27], [15], [25] and [30]). In his
case he dicul ies inc ease as shows [16]. Sp ec al p op e ies o he classical K all p olyno-
mials [28], [29] we e conside ed in [11].
A sp ecial emphasis was gi en o he mo dica ions o classical linea unc ionals (He mi e,
Lague e, Jacobi and Bessel) in he amewo k o he so-called semiclassical o hogonal
p olynomials. Fo example in [27] he Jacobi case wi h wo masses a p oin s
x
=
1 was
conside ed. The hyp e geome ic ep esen a ion o he esul ing p olynomials as well as he
exis ence o a second o de die en ial equa ion ha such p olynomials sa is y ha e b een
es ablished. Also he pa icula cases o he K all p olynomials [28], [29] ha e b een ob ained
om his gene al case as sp ecial cases o limi cases. In [21], [23] (see also [25]) he Lague e
case was conside ed in de ails. In pa icula an inni e o de die en ial equa ion o hese
p olynomials as well as hei ep esen a ion as hyp e geome ic se ies ha e b een ound. The
2
case o mo dica ion o a classical symme ic unc ional (He mi e and Gegenbaue unc ion-
als) was conside ed in [6].
The mo dica ion o classical unc ionals ha e b een conside ed also o he disc e e o hogo-
nal p olynomials. In his di ec ion Ba inck and an Hae ingen [9] ob ained an inni e o de
die ence equa ion o gene alized Meixne p olynomials, i.e., p olynomials o hogonal wi h
esp ec o he mo dica ion o he Meixne weigh wi h a p oin mass a
x
= 0. The same
was ound o gene alized Cha lie p olynomials by Ba inck and Ko eko ek [10]. In a se ies o
pap e s by Al a ez-No da se e . al [2]-[4] he au ho s ha e ob ained he ep esen a ion as hy-
p e geome ic unc ions o gene alized Meixne , Cha lie , K a chuk and Hahn p olynomials
as well as he co esp onding second o de die ence equa ion ha such p olynomials sa is y.
The connec ion o all hese disc e e p olynomials wi h he Jacobi [27] and Lague e [21] yp e
whe e s udied in de ails in [5]. In pa icula , in [5] hey p o ed ha he Jacobi-Ko o nwinde
p olynomials [27] a e a limi case o he gene alized Hahn as well as he Lague e-Ko eko ek
[21], [23] a e o he Meixne ones.
The aim o he p esen con ibu ion is o gi e an unied app oach o his sub jec including
he sp ec al p op e ies by means o he cen al momen s o he p olynomials [12] and he
WKB o semiclassical app oxima ion o he densi y o he dis ibu ion o ze os [8], [42], [43]
and some asymp o ic o mulas o he p olynomials. Also a new in e p e a ion o he K all
p olynomials in e ms o sp ecial Jacobi ma ices will b e gi en.
The plan o he pap e is he ollowing. In Sec ion 2 we gi e a gene al heo y which allows
us o ob ain some gene al o mulas o he K all- yp e p olynomials. F om hese o mulas we
ob ain all he explici o mulas o he ou amilies unde conside a ion, i.e., he Jacobi-
Ko o nwinde [27], he Lague e-Ko eko ek [21], [23], and he He mi e-K all and Gegenbaue -
K all [6]. Also a gene al algo i hm is gi en o gene a e he second o de die en ial equa ions
ha such p olynomials sa is y.
In Sec ion 3 we s udy he sp ec al p op e ies o he Jacobi-Ko o nwinde [27], Lague e-
Ko eko ek [21],[23], He mi e-K all [6] and Gegenbaue -K all [6] p olynomials by means o i s
cen al momen s and he WKB o semiclassical app oxima ion o he densi y o he dis i-
bu ion o ze os. Some pa icula cases a e also included.
Finally, in Sec ion 4 we conside some sp ecial cases o K all- yp e p olynomials ob ained om
he analysis o ce ain yp es o Jacobi ma ices and quad a ic ans o ma ion p olynomial
mappings.
2 The deni ion and he ep esen a ion.
Le
P
n
g
b e a sequence o monic p olynomials o hogonal wi h esp ec o a linea unc ional
L
on he linea space o p olynomials
IP
wi h eal co ecien s dened as (
a; b
can b e
1
,
esp ec i ely)
<
L
; P >
=
Z
b
a
P
(
x
)
(
x
)
dx;
(
x
)
2 C
[
a;b
]
;
(
x
)
>
0 o
x
2
[
a; b
]
:
(1)
Th ough he pap e
IP
will deno e he linea space o p olynomials wi h eal co ecien s.
Le us conside a new sequence
~
P
n
g
o hogonal wi h esp ec o a linea unc ional
U
de-
ned on
IP
ob ained om he ab o e unc ional
L
by adding del a Di ac masses a p oin s
3
x
1
; x
2
; ::; x
m
, i.e.,
<
U
; P >
=
<
L
; P >
+
m
X
i
=1
A
i
P
(
x
i
)
; x
i
2
IR
; A
i
0
:
(2)
We will de e mine he monic p olynomials
~
P
n
g
which a e o hogonal wi h esp ec o he
unc ional
U
and we will p o e ha hey exis o all p osi i e
A
i
. To ob ain his, we can
w i e he Fou ie expansion o
~
P
n
(
x
) in e ms o he p olynomials
P
n
(
x
)
~
P
n
(
x
) =
P
n
(
x
) +
n
1
X
k
=0
a
n;k
P
k
(
x
)
:
(3)
In o de o nd he unknown co ecien s
a
n;k
we will use he o hogonali y o he p olynomials
~
P
n
(
x
) wi h esp ec o
U
, i.e.,
0 =
<
U
;
~
P
n
(
x
)
P
k
(
x
)
>
=
<
L
;
~
P
n
(
x
)
P
k
(
x
)
>
+
m
X
i
=1
A
i
~
P
n
(
x
i
)
P
k
(
x
i
)
;
8
k < n
we nd
a
n;k
=
m
X
i
=1
A
i
~
P
n
(
x
i
)
P
k
(
x
i
)
d
2
k
;
(4)
whe e
d
2
k
=
<
L
;
[
P
k
(
x
)]
2
>
. Finally, he equa ion (3) p o ides us he exp ession
~
P
n
(
x
) =
P
n
(
x
)
m
X
i
=1
A
i
~
P
n
(
x
i
)
n
1
X
k
=0
P
k
(
x
i
)
P
k
(
x
)
d
2
k
=
=
P
n
(
x
)
m
X
i
=1
A
i
~
P
n
(
x
i
)
K e
n
1
(
x; x
i
)
:
(5)
In o de o ob ain he unknown alues
~
P
n
(
x
i
) o each
i
= 1
;
2
; :::; m
, we e alua e (5) in
x
j
; j
= 1
;
2
; :::; m
. In his way, he ob ained linea sys em o equa ions
~
P
n
(
x
j
) +
m
X
i
=1
A
i
~
P
n
(
x
i
)
K e
n
1
(
x
j
; x
i
) =
P
n
(
x
j
)
; j
= 1
;
2
; :::; m;
(6)
has an unique solu ion i and only i he de e minan
1 +
A
1
K e
n
1
(
x
1
; x
1
)
A
2
K e
n
1
(
x
1
; x
2
)
A
m
K e
n
1
(
x
1
; x
m
)
A
1
K e
n
1
(
x
2
; x
1
) 1 +
A
2
K e
n
1
(
x
2
; x
2
)
A
m
K e
n
1
(
x
2
; x
m
)
.
.
.
.
.
.
.
.
.
.
.
.
A
1
K e
n
1
(
x
m
; x
1
)
A
2
K e
n
1
(
x
m
; x
2
)
1 +
A
m
K e
n
1
(
x
m
; x
m
)
(7)
do es no anish o all
n
2
IN
. This is also a necessa y and sucien condi ion o he
exis ence o he
n h
deg ee p olynomial
~
P
n
(
x
) o all
n
2
IN
.
In his wo k we will conside he pa icula cases when we add one o wo del a Di ac masses.
Le us conside hese cases wi h mo e de ails.
4
2.1 The Case o one p oin mass a
x
=
x
1
.
In his case om (5)-(6) we ge
~
P
n
(
x
) =
P
n
(
x
)
A
~
P
n
(
x
1
)
K e
n
1
(
x; x
1
)
;
~
P
n
(
x
1
) =
P
n
(
x
1
)
1 +
A
n
1
X
k
=0
(
P
k
(
x
1
))
2
d
2
k
;
(8)
and he condi ion (7) b ecomes
1 +
A
n
1
X
k
=0
(
P
k
(
x
1
))
2
d
2
k
6
= 0
;
which is always ue o e e y
n
2
IN
since
A
0.
2.2 The Case o wo p oin masses a
x
=
x
1
and
x
2
.
Again we s a om (5)-(6). Then,
~
P
n
(
x
) =
P
n
(
x
)
A
1
~
P
n
(
x
1
)
K e
n
1
(
x; x
1
)
A
2
~
P
n
(
x
2
)
K e
n
1
(
x; x
2
)
;
~
P
n
(
x
1
) =
P
n
(
x
1
)
A
2
K e
n
1
(
x
1
; x
2
)
P
n
(
x
2
) 1 +
A
2
K e
n
1
(
x
2
; x
2
)
1 +
A
1
K e
n
1
(
x
1
; x
1
)
A
2
K e
n
1
(
x
1
; x
2
)
A
1
K e
n
1
(
x
2
; x
1
) 1 +
A
2
K e
n
1
(
x
2
; x
2
)
;
~
P
n
(
x
2
) =
1 +
A
1
K e
n
1
(
x
1
; x
1
)
P
n
(
x
1
)
A
1
K e
n
1
(
x
2
; x
1
)
P
n
(
x
2
)
1 +
A
1
K e
n
1
(
x
1
; x
1
)
A
2
K e
n
1
(
x
1
; x
2
)
A
1
K e
n
1
(
x
2
; x
1
) 1 +
A
2
K e
n
1
(
x
2
; x
2
)
;
(9)
and (7) b ecomes
1 +
A
1
K e
n
1
(
x
1
; x
1
)
A
2
K e
n
1
(
x
1
; x
2
)
A
1
K e
n
1
(
x
2
; x
1
) 1 +
A
2
K e
n
1
(
x
2
; x
2
)
6
= 0
:
Mo eo e , i
A
1
and
A
2
a e nonnega i e cons an s hen he ab o e de e minan is always
p osi i e. To p o e his i is sucien o expand he de e minan and use he Cauchy-Schwa z
inequali y (
P
a
k
b
k
)
2
P
a
2
k
P
b
2
k
.
3 Applica ions o classical p olynomials.
In he p e ious sec ion we conside he p olynomials o hogonal wi h esp ec o a e y gene al
weigh unc ion
(
x
)
2 C
[
a;b
]
;
(
x
)
>
0
; x
2
[
a; b
]. In his sec ion we will conside some pa -
icula cases when
(
x
) is some o he classical weigh unc ions, i.e., he Jacobi, Lague e,
5
He mi e o Gegenbaue weigh unc ions, esp ec i ely. Mo eo e , since in exp essions (8)
and (9) he ke nel p olynomials
K e
n
1
(
x; x
i
) app ea we will conside he case when we add
some del a Di ac masses a he o igin
x
= 0 o a he ends o he in e al o o hogonali y
o he classical p olynomials. The las conside a ion allows us o ob ain explici o mulas o
he ke nel p olynomials in e ms o he classical p olynomials and hei de i a i es [5], [6].
In his way, i we conside he Jacobi case and add wo masses a
x
=
1 we ob ain he well-
known Jacobi-Ko o nwinde p olynomials [27] and o sp ecial alues o he masses
A
1
; A
2
he classical K all p olynomials [28], [29]. Fo Lague e case when
x
= 0 we ob ain he
Lague e-Ko eko ek p olynomials [21], [23]. Finally, o He mi e and Gegenbaue cases when
x
= 0 ( he symme ic case) we ob ained he He mi e-K all and Gegenbaue -K all p olyno-
mials in o duced in [6].
The main da a o he classical p olynomials can b e ound in [17], [36], [39], o he monic
p olynomials see, o ins ance, [5], [6].
3.1 The Jacobi-Ko o nwinde p olynomials.
The Jacobi-Ko o nwinde o hogonal p olynomials we e in o duced by T.H. Ko o nwinde [27].
They can b e ob ained om he gene alized Hahn p olynomials in o duced in [4] as a limi
case [5] and co esp ond o he case o adding wo del a Di ac masses a he ends o he
in e al o o hogonali y o he classical Jacobi p olynomials.
Deni ion 1
The Jacobi-Koo nwinde o hogonal polynomials
P
; ;A;B
n
(
x
)
a e he polyno-
mials o hogonal wi h espec o a linea unc ional
U
on
IP
dened as ol lows (
A; B
0
; >
1
; >
1
)
<
U
; P >
=
Z
1
1
(
+
+ 2)
2
+
+1
(
+ 1)(
+ 1)
(1
x
)
(1 +
x
)
P
(
x
)
dx
+
AP
(1) +
B P
(
1)
:
(10)
Using he exp ession (9) and he p op e ies o classical monic Jacobi Polynomials
P
;
n
(
x
)
we ob ain he ollowing ep esen a ion o
P
; ;A;B
n
(
x
) in e ms o he classical Jacobi p oly-
nomials and hei de i a i es [5], [27]
P
; ;A;B
n
(
x
) =
P
;
n
(
x
) +
n;;
A;B
d
dx
P
1
;
n
(
x
)
n; ;
B ;A
d
dx
P
;
1
n
(
x
)
;
(11)
whe e
n;;
A;B
=
AP
; ;A;B
n
(
1)
;
n
and
n; ;
B ;A
=
B P
B ;A; ;
n
(
1)
;
n
,
P
; ;A;B
n
(
1) and
P
; ;A;B
n
(1) a e gi en by
P
; ;A;B
n
(
1) =
P
;
n
(
1)
B K e
J;;
n
1
(
1
;
1)
P
;
n
(1) 1 +
B K e
J;;
n
1
(1
;
1)
1 +
AK e
J;;
n
1
(
1
;
1)
B K e
J;;
n
1
(
1
;
1)
AK e
J;;
n
1
(
1
;
1) 1 +
B K e
J;;
n
1
(1
;
1)
;
(12)
and
P
; ;A;B
n
(1) = (
1)
n
P
;;B ;A
n
(
1)
:
(13)
The ke nel p olynomials
K e
J
n
1
(
x;
1) a e gi en by
6
K e
J;;
n
1
(
1
;
1) =
(
+
n
+ 1)(
+
+
n
+ 1)(
+ 1)
2
n
1
(
n
1)!(
+ 2)(
+
n
)(
+
+ 2)
;
K e
J;;
n
1
(1
;
1) =
K e
J; ;
n
1
(
1
;
1)
;
K e
J;;
n
1
(
1
;
1) =
(
1)
n
1
(
+
+
n
+ 1)
2
n
1
(
n
1)!
;
(14)
and
;
n
;
;
n
deno e he quan i ies
;
n
=
(
1)
n
1
(2
n
+
+
)(
+ 1)
2
n
1
n
!(
+
n
)(
+ 1)(
+
+ 2)
;
;
n
=
(
1)
n
1
(2
n
+
+
)(
+ 1)
2
n
1
n
!(
+
n
)(
+ 1)(
+
+ 2)
;
(15)
esp ec i ely.
Also he ollowing equi alen ep esen a ion, simila o he ep esen a ion ob ained in [27]
o he monic gene alized p olynomials, is alid
P
; ;A;B
n
(
x
) = (1
nJ
n;;
A;B
nJ
n; ;
B ;A
)
P
;
n
(
x
)+
+[
J
n;;
A;B
(
x
1) +
J
n; ;
B ;A
(1 +
x
)]
d
dx
P
;
n
(
x
)
;
(16)
whe e
J
n;;
A;B
=
AP
; ;A;B
n
(
1) ~
;
n
,
J
n; ;
B ;A
=
B P
B ;A; ;
n
(
1) ~
;
n
and ~
;
n
;
~
;
n
deno e
he quan i ies
~
;
n
=
(
1)
n
(2
n
+
+
+ 1)(
+ 1)
2
n
n
!(
+
n
+ 1)(
+
+ 2)
;
~
;
n
=
(
1)
n
(2
n
+
+
+ 1)(
+ 1)
2
n
n
!(
+
n
+ 1)(
+
+ 2)
:
(17)
F om he ab o e o mula (16) we can ob ain a lo o in e es ing p op e ies, in pa icula
he hyp e geome ic ep esen a ion o he new p olynomials [27], he second o de die en ial
equa ion [27], [20], [5] (see App endix I) and he h ee- e m ecu ence ela ion
x P
; ;A;B
n
(
x
) =
P
; ;A;B
n
+1
(
x
) +
n
P
; ;A;B
n
(
x
) +
n
P
; ;A;B
n
1
(
x
)
; n
0
P
; ;A;B
1
(
x
) = 0
;
and
P
; ;A;B
0
(
x
) = 1
;
(18)
which is a consequence o he o hogonali y o he p olynomials (10). The co ecien s
n
can
b e ob ained equa ing he co ecien s o he
x
n
p owe in (18). Then,
n
=
2
2
(2
n
+
+
)(2
n
+ 2 +
+
)
+
n
(
J
n;;
A;B
J
n; ;
B ;A
)
(
n
+ 1)(
J
n
+1
;;
A;B
J
n
+1
; ;
B ;A
)
:
To ob ain
n
we no ice ha
P
; ;A;B
n
(1)
6
= 0 o all
n
0. Then, om (18)
n
= (1
n
)
P
; ;A;B
n
(1)
P
; ;A;B
n
1
(1)
P
; ;A;B
n
+1
(1)
P
; ;A;B
n
1
(1)
:
7
Also om (16) i is p ossible o ob ain he a io asymp o ics
P
; ;A;B
n
P
;
n
. Fi s ly, we use he
asymp o ic o mula o he (
x
) unc ion [1] o ob ain
J
n;;
A;B
+ 1
n
2
; J
B ;A
n; ;
+ 1
n
2
:
Then, he o mulas o he a io
P
; ;A;B
n
P
;
n
ollow om he classical asymp o ic o mulas
o he a io
1
n
P
0
;
n
(cos
)
P
;
n
(cos
)
in he in e al
2
[
";
"
], 0
< " <<
1 o lo cally uni o mly in
IR
n
[
1
;
1]. They a e ob ained as a simple consequence o he Da b oux o mula in
2
[
";
"
],
0
< " <<
1 (see [39], Theo em 8.21.8, page 196) o he he Da b oux o mula in
IR
n
[
1
;
1]
(see [39], Theo em 8.21.7, page 196), esp ec i ely. F om he ab o e conside a ions we nd
P
; ;A;B
n
(cos
)
P
;
n
(cos
)
= 1
+
+ 2
n
+
(cos
+ 1)(
+ 1) + (cos
1)(
+ 1)
n
2
sin
an [(
n
+
1
2
(
+
+ 1))
1
2
(
+
1
2
)
] +
o
(
1
n
)
;
and
P
; ;A;B
n
(
z
)
P
;
n
(
z
)
= 1
+
+ 2
n
+
2
n
(
z
+ 1)(
+ 1) + (
z
1)(
+ 1)
p
z
2
1
+
o
(
1
n
)
;
alid in
2
[
";
"
], 0
< " <<
1 o he in e al
IR
n
[
1
;
1], esp ec i ely. The las o mula
holds uni o mly in he ex e io o an a bi a y closed cu e which enclose he segmen [
1
;
1],
mo eo e , i
z
2
IR
; z >
1, he igh side exp ession is a eal unc ion o
z
.
3.2 The Lague e-Ko eko ek p olynomials.
The Lague e-Ko eko ek o hogonal p olynomials we e in o duced in [27] as a limi case o
he Jacobi-Ko o nwinde p olynomials and s udied wi h mo e de ails in se e al wo ks [21],
[23], [25]. They also can b e ob ained as a limi case o he gene alized Meixne p olynomials
in o duced in [9], [2] using an app opia e limi ansi ion [5].
Deni ion 2
The Lague e-Koekoek o hogonal polynomials
L
;A
n
(
x
)
a e he polynomials
o hogonal wi h espec o a linea unc ional
U
on
IP
dened as ol lows
<
U
; P >
=
Z
1
0
1
(
+ 1)
x
e
x
P
(
x
)
dx
+
AP
(0)
; A
0
; >
1
:
(19)
Using he algo i hm desc ib ed b e o e (see o mula (8)) we nd o he Lague e-Ko eko ek
p olynomials he ollowing ep esen a ion o mula (see [5], [25] o mo e de ails)
L
;A
n
(
x
) =
L
n
(
x
) +
n
d
dx
L
n
(
x
)
;
n
=
A
(
+ 1)
n
n
!
1 +
A
(
+2)
n
1
(
n
1)!
:
(20)
F om (20) we can ob ain a lo o p op e ies, o example, he hyp e geome ic ep esen a ion
o he new p olynomials [25], he second o de die en ial equa ion [25] (see App endix I) and
he h ee- e m ecu ence ela ion
8
x L
;A
n
(
x
) =
L
;A
n
+1
(
x
) +
n
L
;A
n
(
x
) +
n
L
;A
n
1
(
x
)
; n
0
L
;A
1
(
x
) = 0
;
and
L
;A
0
(
x
) = 1
;
(21)
which is a consequence o he o hogonali y o he p olynomials (19). The co ecien s
n
and
n
a e gi en by o mulas (
L
;A
k
(0)
6
= 0 o all
n
0)
n
= 2
n
+
+ 1 +
n
n
+1
;
n
=
n
L
;A
n
(0)
L
;A
n
1
(0)
L
;A
n
+1
(0)
L
;A
n
1
(0)
:
To ob ain he a io asymp o ics
L
;A
n
L
n
we use he asymp o ic o mula o he (
x
) unc ion
[1] o ob ain
n
+ 1
n
;
and hen om (20) and by using he Pe on Fo mula o he a io
1
p
n
(
L
n
)
0
(
z
)
L
n
(
z
)
o he classical
Lague e p olynomials,
z
2
IC
n
[0
;
1
), (see [40], Eq. (4.2.6) page 133 o [39], Theo em 8.22.3)
we nd
L
;A
n
(
z
)
L
n
(
z
)
= 1 +
+ 1
p
n z
1
1
4
p
n z
(2
+ 1
z
)
+
o
1
n
:
3.3 The He mi e-K all p olynomials.
The He mi e-K all p olynomials we e in o duced in [6]. They can b e ob ained as a quad a ic
ans o ma ion o he Lague e-Ko eko ek p olynomials [6].
Deni ion 3
The
gene alized monic He mi e p olynomials
H
A
n
(
x
)
a e he polynomials o -
hogonal wi h espec o he linea unc ional
U
on
IP
dened as ol lows
<
U
; P >
=
Z
1
1
e
x
2
P
(
x
)
dx
+
AP
(0)
; A
0
:
(22)
Again, om o mula (8) a e some s aigh o wa d calcula ions we ob ain ha he He mi e-
K all p olynomials
H
A
n
(
x
) admi he ollowing ep esen a ions in e ms o he classical p oly-
nomials
H
A
2
m
1
(
x
) =
H
2
m
1
(
x
)
; n
= 2
m
1
; m
= 1
;
2
; :::;
2
xH
A
2
m
(
x
) = 2
xH
2
m
(
x
) +
B
m
d
dx
H
2
m
(
x
)
; n
= 2
m; m
= 0
;
1
;
2
; :::
(23)
B
m
=
A
1 +
A
2(
m
+
1
2
)
(
m
)
(
m
+
1
2
)
m
!
:
No ice ha he o dd p olynomials coincide wi h he classical ones. They a e quad a ic ans-
o ma ions o he Lague e-Ko eko ek p olynomials [6]
H
A
2
m
1
(
x
) =
xL
1
2
m
1
(
x
2
)
; n
= 2
m
1
; m
= 1
;
2
;
3
; :::
H
A
2
m
(
x
) =
L
1
2
;A
m
(
x
2
) =
L
1
2
m
(
x
2
) +
B
m
d
dx
2
L
1
2
m
(
x
2
)
; n
= 2
m; m
= 0
;
1
;
2
; :::
(24)
9
=0
;
=0
w k b
(
x
) =
p
(1 +
m
+
m
2
m x
2
m
2
x
2
)
(1
x
2
)
:
Again, aking he limi lim
n
!1
1
n
=0
;
=0
w k b
(
x
), we nd he known exp ession o he classical
Legend e p olynomials [42]
(
x
) =
1
p
1
x
2
:
WKB Densi y
-1 -0.5 0.5 1
Classical Gegengaue
20000
40000
60000
80000
100000
-1 -0.5 0.5 1
Gegenbaue -Koo nwinde
20000
40000
60000
80000
100000
-1 -0.5 0.5 1
Classical Legend e
20000
40000
60000
80000
100000
-1 -0.5 0.5 1
Legend e-Koo nwinde
20000
40000
60000
80000
100000
Figu e 1: WKB densi y o ze os o
P
A;B ;;
n
(
x
).
In Figu e 1 we ep esen he WKB densi y o ze os o he Legend e-Ko o nwinde and
Gegenbaue -Ko o nwinde (wi h
=
= 5) p olynomials. We ha e plo ed he Densi y
unc ion o die en alues o
n
( om op o b o om)
n
= 10
6
;
10
5
;
10
4
. No ice ha he
alue o he mass do esn' play a c ucial ole, since o
n >>
1
J
n;;
A;B
+1
n
; J
n; ;
B ;A
+1
n
,
indep enden ly o he alues o he masses
A
and
B
.
4.2.2 Lague e-Ko eko ek p olynomials
L
;A
n
(
x
)
.
Again he explici exp ession o
w k b
(
x
) is e y la ge and cumb e some. Fi s ly we can
con ince ou sel es ha using (42) and aking he limi when
A
!
0 we nd
w k bclas
(
x
) =
p
(1
2
+ 2
x
+ 2
x
+ 4
m x
x
2
)
2
x
:
which coincides wi h he classical exp ession [42], [43]. I we now conside he sp ecial case
= 0 we ob ain
=0
w k b
(
x
) =
p
R
(
x
)
2
x
2
n
+
n
2
n
+
x
+
n
x
;
whe e
R
(
x
) =
x
2
2
n
2
n
2
n
5
x
4
n
x
n
2
n x
x
2
n
x
2
2
n
2
n
2
n
x
2
n
x
+
n
2
n x
+
x
2
+
n
x
2
+
+2
x
2
n
+
n
2
n
+
x
+
n
x
( 2
n
2
n
2
n
2
x
4
n
x
4
n
n x
+ 2
n
2
n
2
x
+ 3
x
2
+
+3
n
x
2
+ 2
n x
2
+ 2
n
n x
2
)
:
16
WKB Densi y
246 8 10
Classical Lague e
20
40
60
80
100
246 8 10
Lague e-Koekoek
50
100
150
200
250
300
350
Figu e 2: WKB densi y o ze os o
L
0
;A
n
(
x
).
In Figu e 2 we ep esen he WKB densi y o ze os o he Lague e-Ko eko ek p olynomials
wi h
= 0. We ha e plo ed he densi y unc ion o die en alues o
n
( om op o
b o om)
n
= 10
5
;
5
10
4
;
10
4
;
10
3
. No ice ha he alue o he mass do esn' play a c ucial
ole, since o
n >>
1
n
(
+1)
n
, indep enden ly o
A
.
4.2.3 He mi e-K all p olynomials
H
A
2
m
(
x
)
.
We will analyze only he p olynomials o e en deg ee, i.e.,
~
P
2
m
(
x
). In his case om (40)
and (42)
w k bclas
(
x
) =
p
R
(
x
)
(
B
m
+ 2
B
2
m
m
+ 2
x
2
+ 2
B
m
x
2
)
;
R
(
x
) =
6
B
m
3
B
2
m
+ 24
B
2
m
m
+ 8
B
3
m
m
32
B
3
m
m
2
4
B
4
m
m
2
+ 16
B
4
m
m
3
8
B
m
x
2
9
B
2
m
x
2
32
B
m
m x
2
32
B
2
m
m x
2
+ 4
B
3
m
m x
2
+
+32
B
2
m
m
2
x
2
+ 32
B
3
m
m
2
x
2
4
B
4
m
m
2
x
2
+ 4
x
4
+ 12
B
m
x
4
8
B
2
m
x
4
+
+16
m x
4
+ 32
B
m
m x
4
+ 8
B
2
m
m x
4
8
B
3
m
m x
4
4
x
6
8
B
m
x
6
4
B
2
m
x
6
:
I we ake he limi
A
!
0, again we eco e he classical exp ession [42], [43]
w k b
(
x
) =
p
1 + 4
m
x
2
:
WKB Densi y
-200 -100 100 200
Classical He mi e
20
40
60
80
-200 -100 100 200
He mi e-K all
20
40
60
80
Figu e 3: WKB densi y o ze os o he
H
A
n
(
x
).
In Figu e 3 we ep esen he WKB densi y o ze os o ou gene alized He mi e p olynomials.
We ha e plo ed he Densi y unc ion o die en alues o
n
( om op o b o om)
n
=
2
10
4
;
1
:
5
10
4
;
10
4
;
10
3
. No ice ha he alue o he mass do esn' play a c ucial ole,
since o
n >>
1
B
m
1
2
m
, indep enden ly o
A
.
4.2.4 Gegenbaue -K all p olynomials
G
;A
2
m
(
x
)
.
We will analyze only he p olynomials o e en deg ee, i.e.,
~
P
2
m
(
x
). In his case he exp ession
is e y la ge and we will p o ide only he limi case when
A
!
0 which ag ees wi h he
17
WKB Densi y
-1 -0.5 0.5 1
Classical Gegenbaue
20000
40000
60000
80000
-1 -0.5 0.5 1
Gegenbaue -Koo nwinde
10000
20000
30000
40000
50000
60000
70000
-1 -0.5 0.5 1
Classical Legend e
20000
40000
60000
80000
-1 -0.5 0.5 1
Legend e-Koo nwinde
10000
20000
30000
40000
50000
60000
70000
Figu e 4: WKB densi y o ze os o he
G
;A
n
(
x
).
classical exp ession [42], [43]
w k b
(
x
) =
p
2 + 16
m
2
+ 4
+ 16
m
+
x
2
16
m
2
x
2
16
m x
2
4
2
x
2
2
(1
x
2
)
:
In Figu e 4 we ep esen he WKB densi y o ze os o ou gene alized Gegenbaue p oly-
nomials. No ice ha he alue o he mass do esn' play a c ucial ole, since o
n >>
1,
W
m
1
2
m
2
, indep enden ly o
A
. We ha e plo ed he Densi y unc ion o die en al-
ues o he deg ee o he p olynomials ( om op o b o om)
n
= 2
10
4
;
1
:
5
10
4
;
10
4
;
10
3
o wo die en cases: he gene alized Legend e p olynomials (
=
1
2
) and he gene alized
Gegenbaue wi h
= 5.
5 O he in e es ing examples.
In his sec ion we will gi e some o he examples o amilies o K all- yp e o hogonal p oly-
nomials, ob ained using quad a ic ans o ma ions o he a iable o a gi en sequence o
o hogonal p olynomials. These examples can b e ob ained as an applica ion o he ollowing
heo em [33].
Theo em 2
Le
P
n
g
n
0
be a monic o hogonal polynomial sequence (MOPS) wi h espec
o some uniquely de e mined dis ibu ion unc ion
(
x
)
and le
[
;
]
be he ue in e al o
o hogonali y o
P
n
g
n
0
, wi h
1
< <
+
1
. Le
a
and
be xed eal numbe s,
T
(
x
)
(
x
a
)(
x
b
) +
c
a eal polynomial o deg ee wo and pu
= (
b
a
)
2
4
c
. Le
Q
n
g
n
0
be a sequence o polynomials such ha
Q
2
(
a
) =
; Q
2
n
+1
(
x
) = (
x
a
)
P
n
(
T
(
x
))
o al l
n
= 0
;
1
;
2
;:::
. Assume ha one o he ol lowing condi ions hold
(
i
)
c
+
(
ii
)
c
;
1
<
lim
n
!
+
1
P
n
(
)
P
(1)
n
1
(
)
A
B
P
n
(
)
P
(1)
n
1
(
)
;
whe e
B
+
1
i
= +
1
and
P
(1)
n
g
n
0
deno es he sequence o he associa ed polynomials
o he s kind [14] co esponding o
P
n
g
n
0
. Then,
Q
n
g
n
0
is a MOPS wi h espec o
18
a posi i e deni e linea unc ional i and only i
<
0
; Q
2
n
(
x
) =
P
n
(
T
(
x
))
a
n
(
; c
)
a
n
1
(
; c
)
P
n
1
(
T
(
x
))
hold o al l
n
= 0
;
1
;
2
;:::
and
a
n
(
; c
) =
P
n
(
c
)
P
(1)
n
1
(
c
)
:
In hese condi ions,
Q
n
g
n
0
is o hogonal wi h espec o he uniquely de e mined dis ibu-
ion unc ion
~
(
x
)
dened as
d
~
=
M
(
x
a
)
j
x
a
j
d
(
T
(
x
))
T
0
(
x
)
; <
x
a
+
b
2
< s
whe e
M
=
0
+
F
(
c
;
)
0
;
=
+
4
; s
=
+
4
;
F
(
z
;
) =
Z
1
1
d
(
)
z
is he S iel jes unc ions associa ed o he dis ibu ion unc ion
and
0
=
Z
1
1
d
P
(
x
)
.
5.1 Gene alized He mi e p olynomials wi h a mass a
x
= 0
.
Le
L
n
g
n
0
b e he sequence o he monic classical Lague e p olynomials which a e o hog-
onal wi h esp ec o he weigh unc ion
w
(
x
) =
x
e
x
; x
2
[0
;
1
)
; >
1
:
I
>
0, i
ollows om he las heo em [33] ha , o each
such ha
< <
0, he sequence o
monic p olynomials dened by
Q
2
n
+1
(
x
) =
xL
n
(
x
2
)
; Q
2
n
(
x
) =
L
n
(
x
2
)
a
n
a
n
1
L
n
1
(
x
2
)
;
whe e
a
n
=
L
n
(0)
L
n
1
(1)
(0)
; n
= 0
;
1
;
2
; : : : ;
and
L
n
1
(1)
deno es he asso cia ed p olynomials o he s kind o he Lague e p olyno-
mials is o hogonal wi h esp ec o he measu e
d
(
x
) = (
+ 1)
1 +
0
(
x
)
dx
j
x
j
2
1
e
x
2
dx; x
2
(
1
;
1
)
:
Cho osing
=
we ha e ha , up o a cons an ac o ,
d
(
x
) =
j
x
j
2
e
x
2
dx
, wi h
=
1
2
.
Hence
Q
n
g
n
0
is he sequence o he monic gene alized He mi e p olynomials
Q
n
H
(
)
n
,
>
1
2
(c . [14], page 157). Howe e , i we cho ose
such ha
< <
0, hen one
can see ha he e is always a mass p oin , lo ca ed a
x
= 0. This example gene alizes he
He mi e-K all p olynomials conside ed b e o e.
19
5.2 A ni e 2-p e io dic Jacobi ma ix.
Le
B
n
b e a idiagonal 2-To epli z ma ix, which has he gene al o m
B
n
=
2
6
6
6
6
6
6
6
4
a
1
b
1
000
:::
c
1
a
2
b
2
0 0
:::
0
c
2
a
1
b
1
0
:::
0 0
c
1
a
2
b
2
:::
000
c
2
a
1
:::
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
7
7
7
7
7
7
7
5
2
C
(
n;n
)
(43)
whe e we assume ha
b
1
,
b
2
,
c
1
and
c
2
a e p osi i e numb e s. This sp ecial ma ix has b een
s udied in [19] and also in [32].
Since
b
i
>
0 and
c
i
>
0 o
i
= 1
;
2 hen he e exis s an OPS,
S
n
g
n
0
, such ha
B
n
is
he co esp onding Jacobi ma ix o o de
n
. Le
Q
n
g
n
0
b e he co esp onding monic OPS.
Then
Q
2
n
(
x
) = (
b
1
b
2
)
n
S
2
n
(
x
)
; Q
2
n
+1
(
x
) =
b
1
(
b
1
b
2
)
n
S
2
n
+1
(
x
)
:
Mo eo e , acco ding o [32],
Q
n
g
n
0
can b e ob ained by a quad a ic p olynomial mapping
on a linea ans o ma ion o he monic Chebyshe p olynomials o second kind
U
n
g
. In
ac , pu ing
T
(
x
) = (
x
a
1
)(
x
a
2
)
;
= 2
p
b
1
b
2
c
1
c
2
;
=
b
1
c
1
+
b
2
c
2
;
we ha e
Q
2
n
+1
(
x
) = (
x
a
1
)
P
n
(
T
(
x
))
; Q
2
n
(
x
) =
R
n
(
T
(
x
))
;
whe e
P
n
(
x
) =
n
U
n
x
; R
n
(
x
) =
P
n
(
x
) +
b
2
c
2
P
n
1
(
x
)
o all
n
= 0
;
1
;
2
; : : :
. No ice ha
P
n
is o hogonal wi h esp ec o he dis ibu ion unc ion
P
(
x
) =
U
x
;
supp(
P
) = [
;
+
]
;
whe e
U
is he dis ibu ion unc ion o he Chebyshe p olynomials [17], [39] so ha
d
P
(
x
) =
2
2
p
2
(
x
)
2
dx :
F om [33] he S iel jes unc ion o
Q
n
g
n
0
is
F
Q
(
z
) =
M
a
1
z
+
b
1
c
1
z
a
1
[
F
P
(
T
(
z
))
F
P
(0)]
;
(44)
whe e
F
P
deno es he S iel jes unc ion asso cia ed wi h
P
,
M
=
0
b
1
c
1
F
P
(0) and
0
is
he s momen o
P
. Clea ly,
0
=
R
+
d
P
(
x
) = 1. Fu he mo e, using he S iel jes
unc ion
F
U
o he Chebyshe p olynomials [40] (page 176)
F
P
(
z
) =
1
F
U
z
=
2
2
z
p
(
z
)
2
2
;
whe e he squa e o o is such ha
j
z
+
p
(
z
)
2
2
j
>
whene e
z
62
[
;
+
].
Since 0
62
[
;
+
] o
b
1
c
1
6
=
b
2
c
2
( his is no es ic ion, b ecause he case
b
1
c
1
=
b
2
c
2
20
co esp onds o cons an alues along he diagonal o he co esp onding Jacobi ma ix),
elemen a y compu a ions gi e
F
P
(0) = min
b
1
c
1
; b
2
c
2
g
=b
1
c
1
b
2
c
2
. Hence
M
= 1
min
b
1
c
1
; b
2
c
2
g
b
2
c
2
:
I u ns ou om (44) ha he S iel jes unc ion o
Q
n
g
n
0
eads as
F
Q
(
z
) =
M
a
1
z
1
b
2
c
2
1
z
a
1
1
2
T
(
z
)
p
(
T
(
z
)
)
2
2
+ min
b
1
c
1
; b
2
c
2
g
:
F om his we nd (see [33]) ha
Q
n
g
n
0
is o hogonal wi h esp ec o he dis ibu ion
unc ion
d
Q
(
x
) =
M
(
x
a
1
) +
b
1
c
1
j
x
a
1
j
d
P
(
T
(
x
))
=
M
(
x
a
1
) +
1
2
b
2
c
2
1
j
x
a
1
j
p
4
b
1
b
2
c
1
c
2
(
T
(
x
)
b
1
c
1
b
2
c
2
)
2
dx ;
he supp o b eing he union o wo in e als i
M
= 0 and he union o wo in e als wi h a
singula p oin i
M >
0 ei he
supp(
Q
) =
T
1
(supp(
P
)) i
b
1
c
1
b
2
c
2
;
o
supp(
Q
) =
T
1
(supp(
P
) )
[
a
1
g
i
b
1
c
1
> b
2
c
2
:
We no ice ha
T
1
(supp(
P
) ) =
T
1
([
;
+
])
= [
a
1
+
a
2
2
s;
a
1
+
a
2
2
]
[
[
a
1
+
a
2
2
+
;
a
1
+
a
2
2
+
s
]
;
whe e
=
p
b
1
c
1
p
b
2
c
2
2
+
a
1
a
2
2
2
!
1
=
2
; s
=
p
b
1
c
1
+
p
b
2
c
2
2
+
a
1
a
2
2
2
!
1
=
2
:
As we can see, o he case when
b
1
c
1
> b
2
c
2
a se o p olynomials o hogonal wi h esp ec
o a weigh unc ion, o he o m
(
x
) +
(
x
x
0
) whe e
(
x
) is a con inuous unc ion, i.e.,
a K all- yp e weigh unc ion app ea s in a e y na u al way.
WKB App oxima ion o he dis ibu ion o eigen alues o a idiagonal wo-
p e io dic symme ic ma ix.
To conclude his wo k le us o conside an sp ecial case o a symme ic
n
n
ma ix [7]
H
m
=
0
B
B
B
@
a c
0 0 0
:::
c b d
0 0
:::
0
d a c
0
:::
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
C
C
C
A
:
(45)
Fo his ma ix we will ob ain he densi y o he dis ibu ion o eigen alues, i.e., he WKB
densi y o he co esp onding sequence o o hogonal p olynomials which a e, in gene al, o he
21
K all- yp e. He e we wan o p oin ou ha in he o dd case (
m
= 2
n
+ 1) he co esp onding
p olynomials a e
Q
2
n
+1
(
x
) = (
x
a
)
P
n
(
T
(
x
)), so we can conside only he dis ibu ion o
ze os o
P
n
, since o any
n
,
x
=
a
is always a ze o o he p olynomial and hen an eigen alue
o
H
2
n
. Fu he mo e,
P
n
(
T
(
x
)) is a quad a ic mo dica ion o he Chebyshe p olynomials
U
n
(
x
) and hen hey sa is y a SODE which ollows om he classical ones
(1
x
2
)
U
00
n
(
x
)
3
xU
0
n
(
x
) +
n
(
n
+ 2)
U
n
(
x
) = 0
;
(46)
jus p o iding he change
x
$
T
(
x
). In ac we ha e ha
P
n
(
T
(
x
)) sa ises a SODE (39)
wi h he co ecien s
p
(
x
) =
4
c
2
d
2
(
a b
) +
c
2
+
d
2
+
a x
+
b x
x
2
2
q
(
x
) = 3 (
a
b
+ 2
x
)
(
a b
) +
c
2
+
d
2
+
a x
+
b x
x
2
2
4
c
2
d
2
(
a b
) +
c
2
+
d
2
+
a x
+
b x
x
2
2
(
x
) =
n
(2 +
n
) (
a
b
+ 2
x
)
2
(47)
Fo he e en case he si ua ion is mo e complica ed since we need o calcula e he SODE o
he
R
n
(
T
(
x
)) p olynomials. Using he symb olic p og am
Ma hema ica
[41], as well as he
package
Powe Se ies
de elop ed by Ko ep [26] we ob ain o he
R
n
(
x
) p olynomials a SODE
(39) wi h co ecien s
p
(
x
) =
1 +
n
+
c
4
n
+
c
2
x
+ 2
c
2
n x
1 +
x
2
q
(
x
) =
c
2
+ 2
c
2
n
+ 3
x
+ 3
n x
+ 3
c
4
n x
+ 2
c
2
x
2
+ 4
c
2
n x
2
(
x
) =
2
n
+
c
4
n
3
n
2
n
3
c
4
n
3
c
2
n x
3
c
2
n
2
x
2
c
2
n
3
x
(48)
The change o a iables
x
$
T
(
x
) (
T
(
x
) = (
x
a
)(
x
b
)) in he p e ious SODE yields
~
p
(
x
)
Q
00
2
n
(
x
) + ~
q
(
x
)
Q
0
2
n
(
x
) + ~
(
x
)
Q
2
n
(
x
) = 0
;
whe e he co ecien s a e gi en by
~
p
(
x
) = 4
c
2
d
2
T
00
(
x
)
p
T
(
x
)
a
2
b
2
2
cd
;
~
q
(
x
) = 2
cdT
0
(
x
)
2
q
T
(
x
)
a
2
b
2
2
cd
8
c
2
d
2
p
T
(
x
)
a
2
b
2
2
cd
;
~
(
x
) =
T
0
(
x
)
3
T
(
x
)
a
2
b
2
2
cd
:
(49)
Subs i u ing (47), (49) in (42) we can nd he WKB densi y o he eigen alues o he
Hamil onian ma ix
B
n
. An s aigh o wa d calcula ions show us ha he condi ions o he
Theo em 1 a e sa ised i
n >>
1. The exp ession o he
W K B
(
x
), in b o h cases, is o
la ge and we will only show he ypical b eha iou o he WKB densi y (see gu e 5).
As we can see in gu e 5 we ha e ha all eigen alues a e lo ca ed inside he supp o o he
measu e ecxep he one equal o
a
in he o dd case. In he pic u e he alues
a
= 1
; b
=
2
; c
= 3 and
d
= 4 a e used ( om op o b o om
n
= 10000
;
5000
;
1000
;
100).
22
-4 -2 2 4 6 8
E en Case
500
1000
1500
2000
2500
3000
-4 -2 2 4 6 8
Odd Case
500
1000
1500
2000
2500
3000
Figu e 5: WKB Densi y o he Dis ibu ion o he eigen alues o he symme ic ma ix
H
.
App endix I: The second o de die en ial equa ion o he K all
p olynomials.
In his sec ion we gi e a gene al algo i hm o ob ain he second o de die en ial equa ion
(SODE) which sa is y he conside ed K all p olynomials, deno ed he e by
~
P
n
(
x
), i.e., he
Jacobi-Ko o nwinde , he Lague e-Ko eko ek, he He mi e-K all and Gegenbaue -K all p oly-
nomials. The main ac ha we will use is such ha all o hem can b e ep esen ed in e ms
o he classical amilies
P
n
g
in he o m
q
(
x
)
~
P
n
(
x
) =
a
(
x
)
P
n
(
x
) +
b
(
x
)
P
0
n
(
x
)
;
(50)
whe e
q ; a; b
a e p olynomials in
x
and some unc ion on
n
(see o mulas (16), (20), (23) and
(29)). In he nex able a e ep esen ed
q ; a; b
o each o he amilies
~
P
n
g
~
P
n
(
x
)
q
(
x
)
a
(
x
)
b
(
x
)
P
; ;A;B
n
(
x
) 1 1
nJ
n;;
A;B
nJ
n; ;
B ;A
J
n;;
A;B
(
x
1) +
J
n; ;
B ;A
(1 +
x
)
L
;A
n
(
x
) 1 1
n
H
A
2
m
(
x
) 2
x
2
x B
m
G
;A
2
m
(
x
) 2
x
2
x
(1 +
mW
A
m
)
W
A
m
(1
x
2
)
I is known ha he classical p olynomials sa is y a ce ain SODE o hyp e geome ic yp e
[36], [39] o he o m
(
x
)
d
2
dx
2
P
n
(
x
) +
(
x
)
d
dx
P
n
(
x
) +
P
n
(
x
) = 0
;
(51)
23
whe e deg ee (
)
2
;
deg ee (
)
1
;
deg ee (
) = 0. Now i we ake de i a i es in (50) and
use he SODE (51) we can ob ain o mulas simila o (50) bu o he de i a i es
~
P
0
n
(
x
) and
~
P
00
n
(
x
)
(
x
)
~
P
0
n
(
x
) =
c
(
x
)
P
n
(
x
) +
d
(
x
)
P
0
n
(
x
)
;
s
(
x
)
~
P
00
n
(
x
) =
e
(
x
)
P
n
(
x
) +
(
x
)
P
0
n
(
x
)
;
whe e
; s; c; d; e;
a e some unc ions o
(
x
),
(
x
),
,
q
(
x
)
a
(
x
) and
b
(
x
) dep ending on
x
and
n
( hey a e p olynomials in
x
o b ounded deg ee). The ab o e wo exp essions and (50)
lead o he condi ion
q
(
x
)
~
P
n
(
x
)
a
(
x
)
b
(
x
)
(
x
)
~
P
0
n
(
x
)
c
(
x
)
d
(
x
)
s
(
x
)
~
P
00
n
(
x
)
e
(
x
)
(
x
)
= 0
:
(52)
Expanding he de e minan in (52) by he s column, we nd
~
n
(
x
)
d
2
dx
2
~
P
n
(
x
) + ~
n
(
x
)
d
dx
~
P
n
(
x
) +
~
n
(
x
)
~
P
n
(
x
) = 0
;
(53)
whe e
~
n
(
x
) =
s
(
x
) [
a
(
x
)
d
(
x
)
c
(
x
)
b
(
x
)]
;
~
n
(
x
) =
(
x
)[
e
(
x
)
b
(
x
)
a
(
x
)
(
x
)]
;
~
n
(
x
) =
q
(
x
)[
c
(
x
)
(
x
)
e
(
x
)
d
(
x
)]
:
(54)
In some cases he co ecien s can b e simplied by some ac o and he equa ion (53) b ecomes
mo e simple. To conclude his sec ion we will p o ide he SODE o he ou conside ed p oly-
nomials. We wan o ema k ha in o de o ob ain he explici o mulas o he co ecien s
o he SODE (54) we ha e used he symb olic package
Ma hema ica
[41].
Jacobi-Ko o nwinde p olynomials.
The exis ence o his SODE was p o ed by Ko o nwinde [27] and he co ecien s we e
calcula ed explici ly in [20] and [5]. Using (54) we nd
~
n
(
x
) =
1
x
2
( 1 +
J
n;;
A;B
J
n;;
A;B
+
J
n;;
A;B
+
J
n; ;
B ;A
+
J
n; ;
B ;A
J
n; ;
B ;A
2
J
n;;
A;B
n
+
+2
J
n;;
A;B
2
n
2
J
n; ;
B ;A
n
4
J
n;;
A;B
J
n; ;
B ;A
n
2
J
n;;
A;B
J
n; ;
B ;A
n
2
J
n;;
A;B
J
n; ;
B ;A
n
+
+2
J
n; ;
B ;A
2
n
+ 2
J
n;;
A;B
2
n
2
+ 2
J
n; ;
B ;A
2
n
2
2
J
n;;
A;B
x
2
J
n;;
A;B
x
+ 2
J
n; ;
B ;A
x
+ 2
J
n; ;
B ;A
x
2
J
n;;
A;B
2
n x
2
J
n;;
A;B
J
n; ;
B ;A
n x
+ 2
J
n;;
A;B
J
n; ;
B ;A
n x
+ 2
J
n; ;
B ;A
2
n x
2
J
n;;
A;B
2
n
2
x
+
+2
J
n; ;
B ;A
2
n
2
x
x
2
+
J
n;;
A;B
x
2
+
J
n;;
A;B
x
2
+
J
n;;
A;B
x
2
+
J
n; ;
B ;A
x
2
+
J
n; ;
B ;A
x
2
+
J
n; ;
B ;A
x
2
++2
J
n;;
A;B
n x
2
+ 2
J
n; ;
B ;A
n x
2
)
24
~
n
(
x
) =
+
+ 2
J
n;;
A;B
J
n;;
A;B
+
2
J
n;;
A;B
+ 3
J
n;;
A;B
2
J
n;;
A;B
+
2
J
n;;
A;B
2
J
n; ;
B ;A
3
J
n; ;
B ;A
2
J
n; ;
B ;A
+
J
n; ;
B ;A
+ 2
J
n; ;
B ;A
2
J
n; ;
B ;A
+ 2
J
n;;
A;B
n
2
J
n;;
A;B
n
+
+2
J
n;;
A;B
2
n
2
2
J
n;;
A;B
2
n
+ 2
J
n;;
A;B
2
n
+ 2
J
n; ;
B ;A
n
2
J
n; ;
B ;A
n
+ 6
J
n;;
A;B
J
n; ;
B ;A
n
+
+2
2
J
n;;
A;B
J
n; ;
B ;A
n
6
J
n;;
A;B
J
n; ;
B ;A
n
2
2
J
n;;
A;B
J
n; ;
B ;A
n
2
J
n; ;
B ;A
2
n
2
J
n; ;
B ;A
2
n
+
+2
2
J
n; ;
B ;A
2
n
+ 2
J
n;;
A;B
2
n
2
2
J
n;;
A;B
2
n
2
+ 2
J
n;;
A;B
2
n
2
2
J
n; ;
B ;A
2
n
2
2
J
n; ;
B ;A
2
n
2
+
+2
J
n; ;
B ;A
2
n
2
2
x
x
x
6
J
n;;
A;B
x
+ 3
J
n;;
A;B
x
+
2
J
n;;
A;B
x
9
J
n;;
A;B
x
+ 2
J
n;;
A;B
x
3
2
J
n;;
A;B
x
6
J
n; ;
B ;A
x
9
J
n; ;
B ;A
x
3
2
J
n; ;
B ;A
x
+ 3
J
n; ;
B ;A
x
+ 2
J
n; ;
B ;A
x
+
2
J
n; ;
B ;A
x
+
+4
J
n;;
A;B
n x
+ 2
J
n;;
A;B
n x
+ 2
J
n;;
A;B
n x
8
J
n;;
A;B
2
n x
4
J
n;;
A;B
2
n x
+ 4
J
n; ;
B ;A
n x
+
+2
J
n; ;
B ;A
n x
+ 2
J
n; ;
B ;A
n x
+ 16
J
n;;
A;B
J
n; ;
B ;A
n x
+ 12
J
n;;
A;B
J
n; ;
B ;A
n x
+ 4
2
J
n;;
A;B
J
n; ;
B ;A
n x
+
+12
J
n;;
A;B
J
n; ;
B ;A
n x
+ 4
2
J
n;;
A;B
J
n; ;
B ;A
n x
8
J
n; ;
B ;A
2
n x
4
J
n; ;
B ;A
2
n x
8
J
n;;
A;B
2
n
2
x
4
J
n;;
A;B
2
n
2
x
8
J
n; ;
B ;A
2
n
2
x
4
J
n; ;
B ;A
2
n
2
x
+
x
2
x
2
+ 6
J
n;;
A;B
x
2
+
J
n;;
A;B
x
2
2
J
n;;
A;B
x
2
+ 9
J
n;;
A;B
x
2
+ 2
J
n;;
A;B
x
2
+ 3
2
J
n;;
A;B
x
2
6
J
n; ;
B ;A
x
2
9
J
n; ;
B ;A
x
2
3
2
J
n; ;
B ;A
x
2
J
n; ;
B ;A
x
2
2
J
n; ;
B ;A
x
2
+
2
J
n; ;
B ;A
x
2
2
J
n;;
A;B
n x
2
+ 2
J
n;;
A;B
n x
2
+
+6
J
n;;
A;B
2
n x
2
+ 2
2
J
n;;
A;B
2
n x
2
+ 2
J
n;;
A;B
2
n x
2
2
J
n; ;
B ;A
n x
2
+ 2
J
n; ;
B ;A
n x
2
+
+6
J
n;;
A;B
J
n; ;
B ;A
n x
2
+ 2
2
J
n;;
A;B
J
n; ;
B ;A
n x
2
6
J
n;;
A;B
J
n; ;
B ;A
n x
2
2
2
J
n;;
A;B
J
n; ;
B ;A
n x
2
6
J
n; ;
B ;A
2
n x
2
2
J
n; ;
B ;A
2
n x
2
2
2
J
n; ;
B ;A
2
n x
2
+ 6
J
n;;
A;B
2
n
2
x
2
+ 2
J
n;;
A;B
2
n
2
x
2
+
+2
J
n;;
A;B
2
n
2
x
2
6
J
n; ;
B ;A
2
n
2
x
2
2
J
n; ;
B ;A
2
n
2
x
2
2
J
n; ;
B ;A
2
n
2
x
2
+ 2
x
3
+
x
3
+
x
3
2
J
n;;
A;B
x
3
3
J
n;;
A;B
x
3
2
J
n;;
A;B
x
3
3
J
n;;
A;B
x
3
2
J
n;;
A;B
x
3
2
J
n;;
A;B
x
3
2
J
n; ;
B ;A
x
3
3
J
n; ;
B ;A
x
3
2
J
n; ;
B ;A
x
3
3
J
n; ;
B ;A
x
3
2
J
n; ;
B ;A
x
3
2
J
n; ;
B ;A
x
3
4
J
n;;
A;B
n x
3
2
J
n;;
A;B
n x
3
2
J
n;;
A;B
n x
3
4
J
n; ;
B ;A
n x
3
2
J
n; ;
B ;A
n x
3
2
J
n; ;
B ;A
n x
3
~
n
(
x
) =
n
(1 +
+
+
n
) ( 1 + 3
J
n;;
A;B
J
n;;
A;B
+
J
n;;
A;B
2
J
n;;
A;B
2
+ 3
J
n; ;
B ;A
+
+
J
n; ;
B ;A
J
n; ;
B ;A
+ 8
J
n;;
A;B
J
n; ;
B ;A
+ 2
J
n;;
A;B
J
n; ;
B ;A
+ 2
J
n;;
A;B
J
n; ;
B ;A
2
J
n; ;
B ;A
2
2
J
n;;
A;B
n
2
J
n;;
A;B
2
n
+ 2
J
n;;
A;B
2
n
2
J
n; ;
B ;A
n
8
J
n;;
A;B
J
n; ;
B ;A
n
2
J
n;;
A;B
J
n; ;
B ;A
n
2
J
n;;
A;B
J
n; ;
B ;A
n
2
J
n; ;
B ;A
2
n
+ 2
J
n; ;
B ;A
2
n
+ 2
J
n;;
A;B
2
n
2
+
+2
J
n; ;
B ;A
2
n
2
4
J
n;;
A;B
x
2
J
n;;
A;B
x
+ 2
J
n;;
A;B
2
x
+ 4
J
n; ;
B ;A
x
+ 2
J
n; ;
B ;A
x
+
+2
J
n;;
A;B
J
n; ;
B ;A
x
2
J
n;;
A;B
J
n; ;
B ;A
x
2
J
n; ;
B ;A
2
x
+ 2
J
n;;
A;B
2
n x
2
J
n;;
A;B
2
n x
2
J
n;;
A;B
J
n; ;
B ;A
n x
+ 2
J
n;;
A;B
J
n; ;
B ;A
n x
2
J
n; ;
B ;A
2
n x
+ 2
J
n; ;
B ;A
2
n x
2
J
n;;
A;B
2
n
2
x
+
+2
J
n; ;
B ;A
2
n
2
x
x
2
+
J
n;;
A;B
x
2
+
J
n;;
A;B
x
2
+
J
n;;
A;B
x
2
+
J
n; ;
B ;A
x
2
+
J
n; ;
B ;A
x
2
+
+
J
n; ;
B ;A
x
2
+ 2
J
n;;
A;B
n x
2
+ 2
J
n; ;
B ;A
n x
2
)
Lague e-Ko eko ek p olynomials.
The equa ion o he Lague e-Ko eko ek p olynomials was ound in [25]. F om (54) we ob ain
~
n
(
x
) =
x
n
n
+
n
2
n
+
x
+
n
x
;
~
n
(
x
) =
2
n
3
n
2
n
+ 2
n
2
n
+
n
2
n
+
x
+
x
+
+ 2
n
x
+ 2
n
x
n
2
n x
x
2
n
x
2
;
~
n
(
x
) =
n
2
n
n
n
2
+
n
2
n
+
x
+
n
x
:
He mi e-K all p olynomials.
The equa ion o he He mi e-K all p olynomials was ound in [6]. Using (54) we deduce
~
m
(
x
) =
x
B
m
+ 2
B
2
m
m
+ 2
x
2
+ 2
B
m
x
2
;
~
m
(
x
) = 2
B
m
+ 2
B
2
m
m
+
B
m
x
2
2
B
2
m
m x
2
2
x
4
2
B
m
x
4
;
~
m
(
x
) = 4
m x
3
B
m
2
B
2
m
+ 2
B
2
m
m
+ 2
x
2
+ 2
B
m
x
2
:
(55)
25