WKB APPROXIMATION AND
KRALL-TYPE ORTHOGONAL
POLYNOMIALS .
R.

Al a ez-No da se

, F. Ma cellan
y
Depa amen o de Ma ema icas. Escuela Poli ecnica Sup e io .
Uni e sidad Ca los I I I de Mad id. Bu a que 15, 28911, Leganes, Mad id.
J. Pe onilho
z
Dp o. de Ma ema 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 classica ion:
33C45, 33A65, 42C05.
Abs ac
We gi e an unied 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 die 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 die 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 die en ial equa ion wi h p olynomial co ecien s. The co esp onding measu es
a e gi en in able 1. A die 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 ises.
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 dica ion o a linea unc ional in he linea space
o p olynomials wi h eal co ecien 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 deni e case and Ma cellan and Ma oni [31] o quasi-deni e
linea unc ionals. F om he p oin o iew o die 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 dicul 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 dica 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 die 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 inni e o de die 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 dica ion o a classical symme ic unc ional (He mi e and Gegenbaue unc ion-
als) was conside ed in [6].
The mo dica 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 inni e o de
die ence equa ion o gene alized Meixne p olynomials, i.e., p olynomials o hogonal wi h
esp ec o he mo dica 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 die 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 unied 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 die 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 deni 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 ecien s dened 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 ecien 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 ecien 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 sucien 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 sucien 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.
Deni 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
dened 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 die 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 ecien s

n
can
b e ob ained equa ing he co ecien 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].
Deni 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
dened 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 die 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 ecien 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].
Deni 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
dened 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 die 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 die 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 die 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 die 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 die 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 deni 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
)
dened 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 dened 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 dica 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 ises a SODE (39)
wi h he co ecien 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 ecien 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 ecien 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 ised 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 die 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 die 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 ecien s can b e simplied 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 ecien 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 ecien 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