The distribution of zeros of general q-polynomials
Abstract
A general system of q-orthogonal polynomials is de ned by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analog of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behavior of the coe cients of the recurrence relation. This result is applied to speci c classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah,q-Askey & Wilson, Al Salam-Carlitz and the celebrated q-little and q-big Jacobi.
Full text
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS.
1
R.
Al a ez-No da se
2
Depa amen o de Ma ema icas. Escuela Poli ecnica Supe io .
Uni e sidad Ca los III de Mad id. c/ Bu a que 15, 28911, Leganes, Mad id, Spain and Ins i u o
Ca los I
de Fsica Teo ica y Compu acional Uni e sidad de G anada. E-18071 G anada,
Spain.
E. Buenda
Depa amen o de Fsica Mode na. Facul ad de Ciencias.
Uni e sidad de G anada. E-18075 G anada, Spain.
and
J.S. Dehesa
3
Ins i u o
Ca los I
de Fsica Teo ica y Compu acional and
Depa amen o de Fsica Mode na. Uni e sidad de G anada. E-18071 G anada, Spain.
Key wo ds and ph ases: q-o hogonal p olynomials, h ee- e m ecu ence ela ion, dis ibu ion
o ze os, momen s.
PACS subjec classi ica ion:
02.20.+b ; 02.30.+g ; 03.65.Fd .
AMS (MOS 1991) subjec classi ica ion
33D45, 33E30
Abs ac
A gene al sys em o q-o hogonal p olynomials is dened by means o i s h ee- e m ecu -
ence ela ion. This sys em encompasses many o he known amilies o q-p olynomials, among
hem he q-analog o he classical o hogonal p olynomials. The asymp o ic densi y o ze os
o he sys em is shown o b e a simple and compac exp ession o he pa ame e s which cha -
ac e ize he asymp o ic b eha io o he co ecien s o he ecu ence ela ion. This esul is
applied o sp ecic classes o p olynomials known by he names q-Hahn, q-K a chuk, q-Racah,
q-Askey & Wilson, Al Salam-Ca li z and he celeb a ed q-li le and q-big Jacobi.
1 In o duc ion.
In he las decade an inc easing in e es on he so called q-o hogonal p olynomials (o basic
o hogonal p olynomials) is obse ed ( o a e iew see [1 ], [2] and [3]). The eason is no only
o pu ely in insic na u e bu also b ecause o he so many applica ions in se e al a eas o Ma h-
ema ics ( e.g., con inued ac ions, eule ian se ies, he a unc ions, ellip ic unc ions,...; see o
ins ance [4] and [5 ]) and Physics ( e.g., angula momen um [6 ] and [7 ] and i s q-analog [8]-[11 ],
q-Sh odinge equa ion [12] and q-ha monic oscilla o s [13]-[19 ]). Mo eo e , i is well known he
connec ion b e ween he ep esen a ion heo y o quan um algeb as (Clebsch-Go dan co ecien s,
3j and 6j symb ols) and he q-o hogonal p olynomials, (see [20 ], [21] (Vol. I I I), [22 ], [23 ], [24] ), and
he imp o an ole ha hese q-algeb as play in physical applica ions (see o ins ance [26 ]-[31 ]
and e e ences he ein).
Howe e , he dis ibu ion o ze os o hese p olynomials emains p ac ically unknown o he
b es o ou in o ma ion. The p esen pap e con inues, co ec s and conside ably ex ends he in-
es iga ion o he asymp o ic b eha io o ze os o he q-p olynomials ini ia ed by one o us [32 ].
This is done by he conside a ion o a gene al sys em o q-p olynomials which include mos o he
1
This pap e app ea in he
J. Phys. A: Ma h. Gen.
30
(1997) 6743-6768
2
E-mail add ess [email p o ec ed]; Fax No. +34-1-6249430
3
E-mail add ess [email p o ec ed]; Fax No. +34-58-242862
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
2
q-p olynomials encoun e ed in he li e a u e and hen, s udy o i s dis ibu ion densi y o ze os as
well as he co esp onding asymp o ic limi .
The me ho d o p o o used is e y s aigh o wa d. I is based on an explici o mula o he
momen s-a ound- he-o igin o he disc e e densi y o ze os o a p olynomial wi h a gi en deg ee in
e ms o he co ecien s o he h ee e m ecu ence ela ion [37], as desc ib ed in Lemma 1 gi en
b elow. This me ho d was p e iously employed o no mal (non-q) p olynomials whe e ecu ence
co ecien s a e gi en by means o a a ional unc ion o he deg ee [38], as well as o co esp ond-
ing Jacobi ma ices [39] encoun e ed in quan um mechanical desc ip ion o some physical sys ems.
The pap e is s uc u ed as ollows. Fi s ly, in sec ion 2, one in o duces a gene al se o q-
p olynomials
P
n
(
x
)
q
g
N
n
=0
by means o i s h ee- e m ecu ence ela ion. Sec ion 3 con ains he
main esul s which e e o he disc e e densi y o ze os (i.e. he numb e o ze os p e uni o
ze o in e al) o he p olynomial
P
n
(
x
)
q
,
n
b eing a sucien ly la ge alue, and o i s asymp o ical
limi (i.e., when
n
! 1
). Bo h disc e e and asymp o ic densi ies o ze os a e supp osed o b e
cha ac e ized by he knowledge o all hei momen s. These esul s a e gi en in he o m o ou
heo ems. Theo em 1 gi es he b eha io o he momen s o he disc e e densi y o ze os in e ms
o he pa ame e s dening he ecu ence ela ion. The asymp o ic densi y o ze os is gi en by
Theo ems 2, 3 and 4 in a simila way.
P o o s and de ailed discussion o hese heo ems a e con ained in Sec ions 4 and 5 esp ec i ely.
The u mos eo has b een concen a ed on sea ching o an app opia e asymp o ic densi y o ze os
o ob ain as much in o ma ion as p ossible ab ou he asymp o ic dis ibu ion o ze os o he new
p olynomials. Finally, Sec ion 6 con ains applica ion o heo ems 1, 2, 3 and 4 o mula ed in sec ion
3 o se e al known amilies o q-p olynomials.
2 The gene al sys em o q-o hogonal p olynomials.
The gene al sys em o q-o hogonal p olynomials
P
n
(
x
)
q
g
N
n
=0
is dened by he ecu ence
ela ion
P
n
(
x
) = (
x
a
n
)
P
n
1
(
x
)
b
2
n
1
P
n
2
(
x
)
P
1
(
x
) = 0
; P
0
(
x
) = 1
; n
1
(1)
wi h he co ecien s
a
n
and
b
2
n
1
gi en by
a
n
=
A
X
m
=0
g
m
X
i
=0
(
m
)
i
n
g
m
i
!
q
d
m
n
A
0
X
m
=0
0
@
h
m
X
i
=0
(
m
)
i
n
h
m
i
1
A
q
e
m
n
a
num
n
a
den
n
b
2
n
=
B
X
m
=0
0
@
k
m
X
i
=0
(
m
)
i
n
k
m
i
1
A
q
m
n
B
0
X
m
=0
0
@
l
m
X
i
=0
(
m
)
i
n
l
m
i
1
A
q
s
m
n
(
b
num
n
)
2
(
b
den
n
)
2
(2)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
3
whe e
q
is an a bi a y p osi i e eal numb e bigge han 1. Fu he , he ollowing gene al eque -
imen s on he eal pa ame e s dening
a
n
and
b
2
n
will b e assumed:
1. All memb e s o he sequence
(
m
)
i
; 0
i
h
m
g
A
0
m
=0
,
(
m
)
i
; 0
i
l
m
g
B
0
m
=0
do no anish
simul aneously. So we assu e
a
n
and
b
2
n
no o b e inni e o all
n
.
2. The pa ame e s
(
m
)
i
; 0
i
k
m
g
B
m
=0
and
(
m
)
i
; 0
i
l
m
g
B
0
m
=0
a e such ha
b
2
n
>
0 o
n
1. Then Fa a d's heo em assu es he o hogonali y o he p olynomials
P
n
(
x
)
q
g
N
n
=0
.
3. The ollowing inequali ies a e e ied:
q
d
0
> q
d
1
> : : : > q
d
A
;
q
e
0
> q
e
1
> : : : > q
e
A
0
q
0
> q
1
> : : : > q
B
;
q
s
0
> q
s
1
> : : : > q
s
B
0
(3)
and
g
0
> g
1
> : : : > g
m
;
h
0
> h
1
> : : : > h
m
k
0
> k
1
> : : : > k
m
;
l
0
> l
1
> : : : > l
m
(4)
Condi ions (3) and (4) do no ob iously imply any loss o gene ali y. He e one should also
p oin ou ha he p olynomials discussed in e e ence [32] a e ins ances o he p olynomials
(1)-(2) co esp onding o he alues
g
m
=
k
m
=
h
m
=
e
m
=
l
m
=
s
m
= 0 o all
m
.
3 Main Resul s.
Be o e collec ing he main esul s o his wo k, le us desc ib e Lemma 1 which is he basic o ol
o nd hem.
Lemma 1
Le
P
N
(
x
)
g
be a sys em o polynomials
P
N
(
x
)
dened by he ecu ence ela ion (1),
which is cha ac e ized by he secuences o numbe s
a
n
g
and
b
n
g
. Le he quan i ies
0
=
N ;
0
(
N
)
m
=
Z
b
a
x
m
N
(
x
)
dx; m
= 1
;
2
; :::; N
(5)
be he non-no malized- o-uni y spec al momen s o he polynomials
P
N
(
x
)
, i.e., he momen s
a ound he o igin o he disc e e densi y o ze os
N
(
x
)
, dened by
N
(
x
) =
N
X
i
=1
(
x
x
N ;i
)
;
(6)
x
N ;i
; i
= 1
;
2
; :::; N
g
being he ze os o ha polynomial. I is ull led ha
0
(
N
)
m
=
X
(
m
)
F
(
0
1
;
1
; :::;
j
;
0
j
+1
)
N
X
i
=1
a
0
1
i
b
2
1
i
a
0
2
i
+1
b
2
2
i
+1
: : : b
2
j
i
+
j
1
a
0
j
+1
i
+
j
;
(7)
o
m
= 1
;
2
; :::; N
. The suma ion
X
(
m
)
uns o e al l pa i ions
(
0
1
;
1
; :::;
0
j
+1
)
o he numbe
m
such ha
1.
R
0
+ 2
R
=
m
, whe e
R
and
R
0
deno e he sums
R
=
j
X
i
=1
i
and by
R
0
=
j
1
X
i
=1
0
i
, o
j
1
X
i
=1
0
i
+ 2
j
X
i
=1
i
=
m
(8)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
4
2. I
s
= 0
;
1
< s < j
, hen
k
=
0
k
= 0
o each
k > s
and
3.
j
=
m
2
o
j
=
m
1
2
o m e en o odd espec i ely.
The ac o ial coecien
F
a e dened by
F
(
0
1
;
1
;
0
2
; :::;
0
p
1
;
p
1
;
0
p
) =
m
(
0
1
+
1
1)!
0
1
!
1
!
2
4
p
1
Y
i
=2
(
i
1
+
0
i
+
i
1)!
(
i
1
1)!
i
!
0
i
!
3
5
(
p
1
+
0
p
1)!
(
p
1
1)!
0
p
!
;
(9)
wi h he con en ion
0
=
p
= 1
. Fo he e alua ion o hese coecien s, we mus ake in o
accoun he ol lowing con enc ion
F
(
0
1
;
1
;
0
2
;
2
:::;
0
p
1
;
0
;
0) =
F
(
0
1
;
1
;
0
2
;
2
:::;
0
p
1
)
In (7),
deno es he numbe o non- anishing
i
which a e in ol ed in each pa i ion o
m
.
This Lemma was ini ially ound in a con ex o Jacobi ma ices [37 ]-[38 ]. Jus o unde s and
he p ac ical use o he Lemma, le us gi e he s h ee sp ec al momen s
0
1
=
N
X
i
=1
a
i
;
0
2
=
N
X
i
=1
a
2
i
+ 2
N
1
X
i
=1
b
2
i
;
0
3
=
N
X
i
=1
a
3
i
+ 3
N
1
X
i
=1
b
2
i
(
a
i
+
a
i
+1
)
:
(10)
In he ollowing, he main esul s o his wo k a e collec ed in he o m o ou heo ems. The
s o hem e e s o he disc e e densi y o ze os (6) o he p olynomials dened by (1)-(2) and he
o he h ee conce n wi h he asymp o ic densi y o ze os, i.e., when he deg ee o he p olynomial
ends owa ds inni y. Th oughou he pap e he symb ol
means
beha es as
.
Theo em 1
Le
P
N
(
x
)
q
, e y la ge
N
, be a polynomial dened by he exp essions (1)-(4). The
momen s
0
(
N
)
m
;
m
= 1
;
2
; :::; N
g
o he non-no malized densi y o ze os
N
(
x
) =
P
N
i
=1
(
x
x
N ;i
)
o he polynomial
P
N
(
x
)
q
ha e he ol lowing beha io
1. I
d
0
e
0
=
1
2
(
0
s
0
) = 0
, h ee cases occu :
(a) I
g
0
h
0
>
1
2
(
k
0
l
0
)
. Then
0
(
N
)
m
"
(0)
0
(0)
0
#
m
N
(
g
0
h
0
)
m
+1
:
(11)
(b) I
g
0
h
0
=
1
2
(
k
0
l
0
)
. Then
0
(
N
)
m
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
N
1
2
(
k
0
l
0
)
m
+1
:
(12)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
5
(c) I
g
0
h
0
<
1
2
(
k
0
l
0
)
. Then
0
(
N
)
m
"
(0)
0
(0)
0
#
m
2
N
1
2
(
k
0
l
0
)
m
+1
:
(13)
2. I
d
0
e
0
6
= 0
and/o
0
s
0
6
= 0
, wo cases occu :
(a) i. I
d
0
e
0
<
0
and
0
s
0
<
0
in such a way ha
1
6
= 0
. Then
0
(
N
)
m
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
q
2
(
l n q
)
M
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
d
M
d
M
1
q
1
1
q
1
!
;
(14)
whe e
d
M
d
M
1
deno es he
M
de i a i e wi h esp ec o
1
.
ii. I
d
0
e
0
= 0
and
0
s
0
<
0
and
g
0
h
0
=
k
0
l
0
= 0
. Then
0
(
N
)
m
X
(
m
)
F
(
0
1
;
0
; :::;
0
;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
N :
(15)
iii. I
d
0
e
0
<
0
and
0
s
0
= 0
and
g
0
h
0
=
k
0
l
0
= 0
. Then
0
(
N
)
m
X
(
m
)
F
(0
;
1
; :::;
j
;
0)
"
(0)
0
(0)
0
#
R
N :
(16)
(b) I
d
0
e
0
>
0
and/o
0
s
0
>
0
, h ee die en subcases may occu , namely:
i.
d
0
e
0
>
1
2
(
0
s
0
)
. Then
0
(
N
)
m
"
(0)
0
(0)
0
#
m
q
m
(
N
+1)(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
N
(
g
0
h
0
)
m
:
(17)
ii. I
d
0
e
0
=
1
2
(
0
s
0
)
. Then h ee di e en ypes s il l come up:
A. I
g
0
h
0
>
1
2
(
k
0
l
0
)
, hen
0
(
N
)
m
"
(0)
0
(0)
0
#
m
q
m
(
N
+1)(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
N
(
g
0
h
0
)
m
:
(18)
B. I
g
0
h
0
=
1
2
(
k
0
l
0
)
, hen
0
(
N
)
m
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
q
2
+
m
(
N
+1
)(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
N
m
(
g
0
h
0
)
:
(19)
C. I
g
0
h
0
<
1
2
(
k
0
l
0
)
, hen
0
(
N
)
m
"
(0)
0
(0)
0
#
m
2
q
(
d
0
e
0
)
mN
q
(
d
0
e
0
)
m
1
N
1
2
(
k
0
l
0
)
m
:
(20)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
6
iii.
d
0
e
0
<
1
2
(
0
s
0
)
. Then
0
(
N
)
m
"
(0)
0
(0)
0
#
m
2
q
1
2
(
0
s
0
)
mN
q
1
2
(
0
s
0
)
m
1
N
1
2
(
0
s
0
)
m
:
(21)
The suma ion
X
(
m
)
and he pa ame e
a e as dened in Lemma 1. Besides, he pa ame e s
1
,
2
and
M
a e as ol lows:
1
= [(
d
0
e
0
)
1
2
(
0
s
0
)]
R
0
+
m
2
(
0
s
0
) (22)
2
= (
d
0
e
0
)
j
X
k
=1
k
0
k
+1
+ 2(
0
s
0
)
j
1
X
k
=1
k
k
+1
(23)
M
= [(
g
0
h
0
)
1
2
(
k
0
l
0
)]
R
0
+
m
2
(
k
0
l
0
) (24)
The p o o o his heo em is shown in Sec ion 4.
Theo em 2
Le
P
N
(
x
)
q
be a polynomial dened as in Theo em 1 wi h he adi ional condi ion
(
d
0
e
0
) =
1
2
(
0
s
0
) = 0
. (i.e. case 1)
Le
(
x
)
,
1
(
x
)
and
2
(
x
)
be he asymp o ic (i.e. when
N
! 1
) densi ies o ze os o he
polynomial
P
N
(
x
)
q
dened by
(
x
) = lim
N
!1
N
(
x
)
;
1
(
x
) = lim
N
!1
1
N
N
x
N
(
g
0
h
0
)
;
2
(
x
) = lim
N
!1
1
N
N
x
N
1
2
(
k
0
l
0
)
(25)
and hei co esponding momen s a e as ol lows:
0
m
= lim
N
!1
0
(
N
)
m
;
m
(1) = lim
N
!1
0
(
N
)
m
N
(
g
0
h
0
)
m
;
m
(2) = lim
N
!1
0
(
N
)
m
N
(
k
0
l
0
)
m
2
(26)
o
m
= 0
;
1
;
2
; :::
espec i ely. He e
N
(
x
)
deno es he (disc e e) densi y o ze os o he polynomial
P
N
(
x
)
q
. I u ns ou ha
0
m
=
1
; m
0 (27)
and
1. I
g
0
h
0
>
1
2
(
k
0
l
0
)
. Then
m
(1) =
"
(0)
0
(0)
0
#
m
; m
0 (28)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
7
2. I
g
0
h
0
=
1
2
(
k
0
l
0
)
. Then
m
(2) =
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
; m
0 (29)
3. I
g
0
h
0
<
1
2
(
k
0
l
0
)
. Then
m
(2) =
"
(0)
0
(0)
0
#
m
2
; m
0
:
(30)
He e he coecien s
F
and he symbol o summa ion
X
(
m
)
a e as in Theo em 1.
Theo em 3
Le
P
N
(
x
)
q
be a polynomial dened as in Theo em 1 wi h he adi ional condi ion
(
d
0
e
0
)
0
and
1
2
(
0
s
0
)
0
. (i.e. subcase 2a)
Le
(
x
)
and
1
(
x
)
be he asymp o ic densi ies o ze os o he polynomial
P
N
(
x
)
q
dened by
(
x
) = lim
N
!1
N
(
x
);
1
(
x
) = lim
N
!1
1
N
N
(
x
)
(31)
and hei co esponding momen s a e as ol lows:
0
m
= lim
N
!1
0
(
N
)
m
;
0
m
(1) = lim
N
!1
0
(
N
)
m
N
(32)
o
m
0
, espec i ely. I u ns ou ha :
1. I
d
0
e
0
<
0
and
0
s
0
<
0
in such a way ha
1
6
= 0
. Then
0
m
=
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
q
2
(
l n q
)
M
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
d
M
d
M
1
q
1
1
q
1
!
;
(33)
and
0
0
(1) = 1
;
0
m
(1) = 0
; m
1
:
(34)
2. I
d
0
e
0
= 0
and
0
s
0
<
0
and
g
0
h
0
=
k
0
l
0
= 0
. Then
0
m
=
1
; m
0 (35)
0
m
(1) =
8
>
>
>
>
>
<
>
>
>
>
>
:
1
m
= 0
X
(
m
)
F
(
0
1
;
0
; :::;
0
;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
m
1
:
(36)
3. I
d
0
e
0
<
0
and
0
s
0
= 0
and
g
0
h
0
=
k
0
l
0
= 0
. Then
0
m
=
1
; m
0 (37)
0
m
(1) =
8
>
>
>
>
<
>
>
>
>
:
1
m
= 0
X
(
m
)
F
(0
;
1
;
0
; :::;
j
;
0)
"
(0)
0
(0)
0
#
R
m
1
:
(38)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
8
He e he coecien s
F
and he symbol o summa ion
X
(
m
)
and he pa ame e s
1
,
2
and
M
a e
as in Theo em 1.
Theo em 4
Le
P
N
(
x
)
q
be a polynomial dened as in Theo em 1 wi h he adi ional condi ion
(
d
0
e
0
)
>
0
and/o
1
2
(
0
s
0
)
>
0
. (i.e. subcase 2b)
Le
(
x
)
,
1
(
x
)
,
2
(
x
)
,
3
(
x
)
,
++
1
(
x
)
,
++
2
(
x
)
and
++
3
(
x
)
be he asymp o ic densi ies o
ze os o he polynomial
P
N
(
x
)
q
gi en by
(
x
) = lim
N
!1
N
(
x
)
;
(39)
1
(
x
) = lim
N
!1
N
xq
(
d
0
e
0
)
N
N
(
g
0
h
0
)
!
;
2
(
x
) = lim
N
!1
N
xq
(
d
0
e
0
)
N
N
1
2
(
k
0
l
0
)
!
;
3
(
x
) = lim
N
!1
N
xq
1
2
(
0
s
0
)
N
N
1
2
(
k
0
l
0
)
!
;
(40)
++
1
(
x
) = lim
N
!1
(
m
)
q
(
mN
)
q
N
xq
(
d
0
e
0
1)
N
N
(
g
0
h
0
)
!
;
++
2
(
x
) = lim
N
!1
(
m
)
q
(
mN
)
q
N
xq
(
d
0
e
0
1)
N
N
1
2
(
k
0
l
0
)
!
;
++
3
(
x
) = lim
N
!1
(
m
)
q
(
mN
)
q
N
xq
1
2
(
0
s
0
2)
N
N
1
2
(
k
0
l
0
)
!
;
(41)
and hei co esponding momen s a e as ol lows:
0
m
= lim
N
!1
0
(
N
)
m
(42)
m
(1) = lim
N
!1
0
(
N
)
m
N
(
g
0
h
0
)
q
(
d
0
e
0
)
mN
m
(2) = lim
N
!1
0
(
N
)
m
N
1
2
(
k
0
l
0
)
q
(
d
0
e
0
)
mN
m
(3) = lim
N
!1
0
(
N
)
m
N
1
2
(
k
0
l
0
)
q
1
2
(
0
s
0
)
mN
(43)
++
m
(1) = lim
N
!1
(
m
)
q
(
mN
)
q
0
(
N
)
m
N
(
g
0
h
0
)
q
(
d
0
e
0
1)
mN
++
m
(2) = lim
N
!1
(
m
)
q
(
mN
)
q
0
(
N
)
m
N
1
2
(
k
0
l
0
)
q
(
d
0
e
0
1)
mN
++
m
(3) = lim
N
!1
(
m
)
q
(
mN
)
q
0
(
N
)
m
N
1
2
(
k
0
l
0
)
q
1
2
(
0
s
0
2)
mN
(44)
o
m
0
, espec i ely, and whe e symbol
(
n
)
q
deno es he
q-basic numb e
(
n
)
q
=
q
n
1
q
1
;
(45)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
9
ela ed wi h he q-numbe s
[
n
]
q
=
q
n
q
n
q
q
1
by o mula
(
n
)
q
=
q
n
1
2
[
n
]
q
1
2
. I u ns ou ha
0
m
=
1
; m
0 (46)
and
1.
d
0
e
0
>
1
2
(
0
s
0
)
. Then
m
(1) =
8
>
>
>
>
<
>
>
>
>
:
1
m
= 0
"
(0)
0
(0)
0
#
m
q
m
(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
m
1
:
(47)
Also,
++
m
(1) =
8
>
<
>
:
1
m
= 0
(
q
m
1)
m
(1)
m
1
(48)
2. I
d
0
e
0
=
1
2
(
0
s
0
)
. Then h ee di e en si ua ion come up:
(a) I
g
0
h
0
>
1
2
(
k
0
l
0
)
. Then he momen s
m
(1)
and
++
m
(1)
ha e he same alues as
in he p e ious case., i.e., as o mulas (47) and (48).
(b) I
g
0
h
0
=
1
2
(
k
0
l
0
)
, hen
m
(1) =
8
>
>
>
>
>
<
>
>
>
>
>
:
1
m
= 0
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
q
2
+
m
(1
)(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
m
1
(49)
Also,
++
m
(1) =
8
>
<
>
:
1
m
= 0
(
q
m
1)
m
(1)
m
1
:
(50)
(c) I
g
0
h
0
<
1
2
(
k
0
l
0
)
, hen
m
(2) =
8
>
>
>
>
<
>
>
>
>
:
1
m
= 0
"
(0)
0
(0)
0
#
m
2
1
q
(
d
0
e
0
)
m
1
m
1
(51)
Also,
++
m
(2) =
8
>
<
>
:
1
m
= 0
(
q
m
1)
m
(2)
m
1
(52)
3.
d
0
e
0
<
1
2
(
0
s
0
)
. Then
m
(3) =
8
>
>
>
>
<
>
>
>
>
:
1
m
= 0
"
(0)
0
(0)
0
#
m
2
1
q
1
2
(
0
s
0
)
m
1
m
1
(53)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
16
hen he dominan e m in he
(m)-summa ion
o he exp ession (71) is he one
co esp onding o he pa i ion (0
; m;
0
; :::;
0). The e o e
R
0
= 0,
R
=
m
2
,
= 1,
2
= 0,
M
=
1
2
(
k
0
l
0
) and
0
(
N
)
m
F
(0
; m;
0
; :::;
0)
"
(0)
0
(0)
0
#
m
2
q
1
2
(
0
s
0
)
mN
q
1
2
(
0
s
0
)
m
1
N
1
2
(
0
s
0
)
m
;
which coincides wi h (21) since
F
(0
; m;
0
; :::;
0) = 1.
This comple ely p o es he Theo em 1.
As a conclusion o his sec ion we p o ide he scheme wi h all die en p osibili ies ob ained
in his sec ion.
Scheme:
The ca ac e iza ion o gene al q-p olynomials by i s sp ec al p op e ies.
1
:
d
0
e
0
=
1
2
(
0
s
0
)
8
>
<
>
:
(
a
)
g
0
h
0
>
1
2
(
k
0
l
0
)
(
b
)
g
0
h
0
=
1
2
(
k
0
l
0
)
(
c
)
g
0
h
0
<
1
2
(
k
0
l
0
)
2
:
d
0
e
0
6
= 0
0
s
0
6
= 0
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(
a
)
d
0
e
0
0
0
s
0
0
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(
i
)
(
d
0
e
0
<
0
0
s
0
<
0
1
6
= 0
(
ii
)
(
d
0
e
0
= 0
0
s
0
<
0
g
0
h
0
=
k
0
l
0
= 0
(
iii
)
(
d
0
e
0
<
0
0
s
0
= 0
g
0
h
0
=
k
0
l
0
= 0
(
b
)
d
0
e
0
>
0
and=o
0
s
0
>
0
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
(
i
)
d
0
e
0
>
1
2
(
0
s
0
)
(
ii
)
d
0
e
0
=
1
2
(
0
s
0
)
8
>
<
>
:
A
)
g
0
h
0
>
1
2
(
k
0
l
0
)
B
)
g
0
h
0
=
1
2
(
k
0
l
0
)
C
)
g
0
h
0
<
1
2
(
k
0
l
0
)
(
iii
)
d
0
e
0
<
1
2
(
0
s
0
)
5 Sea ching o a no malized densi y o ze os.
In his Sec ion he asymp o ic dis ibu ion o ze os o he p olynomial
P
N
(
x
)
q
dened by Eqs.
(1)-(4) will b e discussed. In pa icula Theo ems 2-4 will b e p o ed. The s a ing p oin will b e
Theo em 1.
F om Theo em 1, one obse es ha he momen s
0
(
N
)
m
o he (non-no malized) densi y o
ze os
N
(
x
) dep ends on
N
as ollows:
N
am
+1
in case 1
;
C ons an
in sub case 2(a)i
;
N
in sub cases 2(a)ii-2(a)iii
;
N
am
q
bmN
in case 2b
;
(72)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
17
whe e he cons an s
a
and
b
a e known and dis inc o each case. Ob iously we would like o ha e
a no malized densi y o ze os
no m
N
(
x
). The usual way o ha e i is o imp ose ha he
momen
o o de ze o
b e equal uni y, wha p e mi s o w i e
no m
N
(
x
) =
1
N
N
(
x
)
;
(73)
whose momen s
~
0
(
N
)
m
will b e ela ed o hose o
N
(
x
) by
~
0
(
N
)
m
=
1
N
0
(
N
)
m
; m
0
:
(74)
Then, om (72) and (74) i is clea ha he
N
-dep endence o he momen s o he
no malized o
uni y
densi y o ze os is gi en by
N
am
in case 1
;
N
1
in sub case 2(a)i
;
C ons an
in sub cases 2(a)ii-2(a)iii
;
N
am
1
q
bmN
in case 2b
;
(75)
As said b e o e, we a e in e es ed in he asymp o ic densi y o ze os. I his is dened by
(
x
) = lim
N
!1
N
(
x
)
;
(76)
hen aking in o accoun ha
0
(
N
)
m
ha e a
N
-dep endence o he o m (72), i s momen s
0
m
gi en
by
0
m
= lim
N
!1
0
(
N
)
m
will b e inni y in case 1, sub cases 2(a)ii and 2(a)iii and in case 2b; and cons an gi en by (14)
in sub case 2(a)i. The e o e, he exp essions (27), (33), (35) and (37) o heo ems 2, 3 and 4,
esp ec i ely, ha e b een p o ed.
I one wan s o ha e some in o ma ion ab ou he asymp o ic dis ibu ion o ze os in case 1,
sub cases 2(a)ii and 2(a)iii and in case 2b, one needs o in o duce a no maliza ion ac o and/o
a scaling ac o in o he densi y
N
(
x
) in he sense disscused in Eq. (55) and (56). Le us s
hink o a
scaled
densi y. Fo he case 1 he e is no scaling ac o
D
which leads o an asymp o ic
densi y o ze os whose momen s ha e non-ze o, ni e alues unless he scaling ac o b e o he
o m
D
=
N
a
1
m
bu his is no use ul since i would oblige o dene a die en
scaled
asymp o ic
densi y unc ion o each momen . Con a y o his, o he case 2b one can conside scaling ac o
D
=
N
a
q
bN
and dene he disc e e densi y o ze os gi en by
N
(
x
) =
N
x
q
bN
N
a
and he asymp o ic densi y o ze os gi en by
(
x
) = lim
N
!1
N
x
q
bN
N
a
(77)
whowe momen s
m
a e aco ding o (56), as ollows
m
= lim
N
!1
0
(
N
)
m
q
mbN
N
am
:
(78)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
18
F om (72) and (78), i is clea ha all he quan i ies
m
ha e ni e alues. I is only missing o
ake he pa ame e s
a
and
b
o he die en sub cases o 2b.
Fo he sub cases 2(b)i, 2(b)iiA and 2(b)iiB i u ns ou ha
a
=
g
0
h
0
and
b
=
d
0
e
0
.
Then, as in exp ession (77), one can dene he asymp o ic densi y unc ion
1
(
x
) in he o m
1
(
x
) = lim
N
!1
N
xq
(
d
0
e
0
)
N
N
(
g
0
h
0
)
!
;
(79)
whose momen s
m
(1) gi en by
m
(1) = lim
N
!1
0
(
N
)
m
N
(
g
0
h
0
)
m
q
(
d
0
e
0
)
mN
;
(80)
ha e, acco ding o (17) and (18), he alues ( o
m
1)
m
(1) =
"
(0)
0
(0)
0
#
m
q
m
(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
(81)
in he sub cases 2(b)i and 2(b)iiA, and, acco ding o (19), he alues
m
(1) =
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
q
2
+
m
(1
)(
d
0
e
0
)
q
m
(
d
0
e
0
)
1
(82)
in he sub case 2(b)iiB. Rema k ha he exp essions (81) and (82) a e iden ical o (47) and (49)
o Theo em 4, esp ec i ely. Simila y, o he sub cases 2(b)iiC i u ns ou ha
a
=
1
2
(
k
0
l
0
)
and
b
=
d
0
e
0
. Then, as in exp ession (77), one denes he asymp o ic densi y unc ion
2
(
x
)
by (40), whose momen s
m
(2) gi en by (43) ha e, acco ding o (20), he alues gi en by (51).
Finally, o he sub case 2(b)iii one has he densi y
3
(
x
) dened by (40), whose momen s
m
(3),
gi en by (43), ha e acco ding o (21), he alues gi en by (53). Fo he en i e case 2b i happ ens
ha , acco ding o (78) and since
0
(
N
)
0
=
N
,
0
=
0
(1) =
0
(2) =
0
(3) =
1
;
as in Theo em 4 is also p oin ed ou .
Le us now sea shed o a
no malized o uni y
asymp o ic densi y o ze os. The simples way
is o dene i as
1
(
x
) = lim
N
!1
no m
N
(
x
) = lim
N
!1
1
N
N
(
x
)
;
(83)
whe e he Eq. (73) has b een used. I s momen s gi en by
0
0
(1) = 1
;
0
m
(1) = lim
N
!1
1
N
0
(
N
)
m
; m
1
;
(84)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
19
ha e, aking in o accoun (75), he ollowing alues
0
0
(1) = 1
0
m
(1) =
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
1
m
1 in cases 1 and 2b
0
m
1 sub case 2(a)i
X
(
m
)
F
(
0
1
;
0
; :::;
0
;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
m
1 sub case 2(a)ii
X
(
m
)
F
(0
;
1
;
0
; :::;
j
;
0)
"
(0)
0
(0)
0
#
R
m
1 sub case 2(a)iii
(85)
Then exp essions (34)-(38) o Theo em 3 has b een demons a ed. So, Theo em 3 has en i ely
p o ed.
Fo he case 1 and sub case 2b one would like o ha e in o ma ion mo e use ul han ha
exp essed by (85), keeping he
no maliza ion o uni y
o he densi y
1
(
x
) gi en by (83). The e o e
one has o
comp ess
he sp ec um o ze os by in o ducing a scaling ac o . In he case 1 i is e y
easy o nd ha ac o by lo oking a he exp ession (75): i is
D
=
N
a
. Then one denes om
(75) and (83) he densi y unc ion
(
x
) = lim
N
!1
no m
N
x
N
a
= lim
N
!1
1
N
N
x
N
a
(86)
whose momen s a e acco ding o (56) and (84) as
0
= 1
;
m
= lim
N
!1
0
(
N
)
m
N
am
+1
; m
1
:
(87)
F om (72) and (87) i is ob ious ha he quan i ies
m
ha e ni e alues. One has only o ake
he alues o
a
in he die en sub cases o he case 1. Fo he sub case 11,
a
=
g
0
h
0
; hen he e
i is con enien o dene, acco ding o (86), he ollowing asymp o ic densi y o ze os
1
(
x
) = lim
N
!1
1
N
N
x
N
g
0
h
0
;
whose momen s a e, acco ding o (87) and (11), as ollows
0
(1) = 1;
m
(1) =
"
(0)
0
(0)
0
#
m
; m
1
;
which is he exp ession (28) o Theo em 2.
Fo he sub cases 12 and 13, i u ns ou ha
a
=
1
2
(
k
0
l
0
), which denes he ollowing
asymp o ic densi y o ze os
2
(
x
) = lim
N
!1
1
N
N
x
N
1
2
(
k
0
l
0
)
;
whose momen s ha e, acco ding o (87) and (12), he alues (
0
(2) = 1)
m
(2) =
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
"
(0)
0
(0)
0
#
R
0
"
(0)
0
(0)
0
#
R
; m
1
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
20
o he sub case 12, and, acco ding o (87) and (13), he alues
0
(2) = 1;
m
(2) =
"
(0)
0
(0)
0
#
m
2
; m
1
;
o he sub case 13. Rema ks ha he las wo exp essions coincide wi h he exp essions (29) and
(30) o Theo em 2, esp ec i ely. Then his Theo em has b een en i ely p o ed.
Fo he sub case 2b he scaled
no maliza ion o uni y
asymp o ic densi y unc ion o he o m
(86) would also ha e all i s momen s o o de o he han ze o equal o inni e. No o he scaling
ac o would b e able o make ni e hese momen s unless
D
=
N
a
+
1
m
q
mN
, bu his ac o is
o use ulness o easons al eady discussed. The e o e one is obliged o change he no maliza ion
ac o in his sub case. He e he disc e e densi y o ze os
+
N
(
x
) is no malized so ha i s momen s
a e dened by
+(
N
)
m
=
q
m
1
q
mN
1
0
(
N
)
m
; m
0
;
i.e., ha
+
N
(
x
) =
(
m
)
q
(
mN
)
q
N
(
x
) (88)
when (
m
)
q
and (
mN
)
q
a e q-numb e s dened by Eq.(45). This no maliza ion ac o has he
ollowing ele an p op e y: I ends o
N
1
i
m
!
0 and
q
!
1. In pa icula , his implies ha
+(
N
)
0
= 1
:
Fu he mo e, o he case 2b unde conside a ion i u ns ou ha he
N
dep endence o
+(
N
)
m
is
as
N
am
q
(
b
1)
mN
. his dep endence sugges o analize he asymp o ic sp ec um o ze os by means
o he asymp o ic densi y unc ion dened by
++
(
x
) = lim
N
!1
N
x
N
a
q
(
b
1)
N
;
(89)
whose momen s
++
m
a e gi en by
++
m
= lim
N
!1
+(
N
)
m
N
am
q
(
b
1)
mN
= lim
N
!1
(
q
m
1)
0
(
N
)
m
(
q
mN
1)
N
am
q
(
b
1)
mN
:
(90)
Taking in o accoun his exp ession oge he wi h he alues (17)-(21) o
0
(
N
)
m
gi en in he Theo em
1, one obse es ha o he sub cases 2(b)i, 2(b)iiA and 2(b)iiB he pa ame e
a
and
b
ake he
alues
a
=
g
0
h
0
; b
=
d
0
e
0
and he app opia e asymp o ic densi y o ze os is, acco ding o (88)-(89), he unc ion
++
1
(
x
)
gi en by (41) in Theo em 4.
Fo he sub case 2(b)iiC i u ns ou ha
a
=
1
2
(
k
0
l
0
)
; b
=
d
0
e
0
:
Then, he app opia e asymp o ic densi y o ze os o his sub case is, acco ding o (88)-(89), he
unc ion
++
2
(
x
) gi en by (41) in Theo em 4.
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
21
Finally o he case 2(b)iii
a
=
1
2
(
k
0
l
0
),
b
=
1
2
(
0
s
0
) and he app opia e asymp o ic densi y
o ze os is, acco ding o (88)-(89), he unc ion
++
3
(
x
) gi en by (41) in Theo em 4.
Now he Eq. (90) and he alues (17)-(21) o
0
(
N
)
m
gi es in a s aigh o wa d manne he
momen s
++
m
(1),
++
m
(2) and
++
m
(3) o he asymp o ic densi y unc ions
++
1
(
x
),
++
2
(
x
) and
++
3
(
x
). Indeed, he alues o hese quan i ies a e gi en by he Eqs. (48) o he sub cases 2(b)i,
2(b)iiA, (50) o he sub case 2(b)iiB, (52) o he sub case 2(b)iiC and (54) o he sub case 2(b)iii,
esp ec i ely. This en i ely p o es Theo ems 2-4.
6 Applica ions.
In his Sec ion we will use he heo ems ob ained in he wo p e ious sec ions o in es iga e
he sp ec al p op e ies o se e al known amilies o o hogonal q-p olynomials. Le us make he
obse a ion ha o a ni e p olynomial sequence (e.g. Hahn, Racah and K a chuk p olynomials),
i.e., when he deg ee
n
o he p olynomial is b ounded by a xed pa ame e
N
(no o b e con used
wi h he same le e p e iously used as gene ic deg ee o p olynomials), i is assumed ha
N
is
sucien ly la ge and 1
<< n
N
so ha Eq. (61) b e ullled.
The q-Hahn p olynomials
h
;
n
(
q
x
; N
)
.
The q-Hahn p olynomials
h
;
n
(
q
x
; N
) play a undamen al ole in he Rep esen a ion Theo y o
he q-Algeb as
S U
q
(2) and
S U
q
(1
;
1) (see [20], [22], [21 ]). They also app ea in nume ous physical
applica ions since e.g. he Clebsh-Go dan Co ecien s o he q-Algeb as
S U
q
(2) and
S U
q
(1
;
1) a e
p op o ional o hem. The heo y and applica ions o he q-Hahn and classical Hahn p olynomials
ha e some close pa allels. So, e.g. q-Hahn and classical Hahn p olynomials app ea s in he analysis
o unc ions on he la ice o subspaces o a ni e ec o space and he la ice o subse s o a ni e
se , esp ec i ely. These p olynomials e i y he ecu ence ela ion [3] (page 59)
h
;
n
(
q
x
; N
) = [
q
x
(1
A
n
1
C
n
1
)]
h
;
n
1
(
q
x
; N
) +
B
n
1
h
;
n
2
(
q
x
; N
)
;
(91)
whe e
B
n
=
A
n
1
C
n
, and he A and C pa ame e s a e
A
n
=
1
q
1+
n
1
q
1+
n
1
q
N
+
n
(1
q
1+2
n
) (1
q
2+2
n
)
;
C
n
=
q
n
(1
q
n
) ( 1
q
n
)
q
N
q
1+
n
(1
q
2
n
) (1
q
1+2
n
)
:
Le us also p oin ou ha
n
+1
A
n
=
n
(92)
whe e
n
is he leading co ecien o he p olynomial. The compa ison o Eqs. (91) and (1) gi es
ha
a
num
n
=
(0)
0
q
3
n
=
2
(1 +
)
q
N
+1
q
3
n
; a
den
n
=
(0)
0
q
4
n
=
2
2
q
N
q
4
n
:
and
(
b
num
n
)
2
=
(0)
0
q
7
n
=
4
3
q
N
q
7
n
;
(
b
den
n
)
2
=
(0)
0
q
8
n
=
4
4
q
8
n
:
Then,
g
m
=
h
m
=
k
m
=
l
m
= 0 o all
m
= 0
;
1
; ::N
and
d
0
= 3
; e
0
= 4
;
0
= 7
; s
0
= 8
:
This is he case
d
0
e
0
<
0 and
0
s
0
<
0, i.e., case 2(a)i. The e o e, Eqs. (33) and (34) o
Theo em 3 gi e us he momen s
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
22
0
m
=
X
(
m
)
F
(
0
1
;
1
; :::;
j
;
0
j
+1
)
q
P
j
k
=1
k
0
k
+1
2
P
j
1
k
=1
k
q
(1 +
)
R
0
q
N
(
q
+
q
1
)
R
1
q
m
2
1
(93)
o he asymp o ic densi y o ze os
(
x
) dened by Eq. (31), and
0
m
(1) =
8
>
<
>
:
1
m
= 0
0
m
1
(94)
o he co esp onding asymp o ic quan i y
1
(
x
) gi en by Eq. (31).
q-K a chuk p olynomials
k
p
n
(
q
x
; N
)
.
The ma ix elemen s o he ep esen a ions
T
l
o he
U
q
(
sl
2
) quan um algeb a a e p op o ional
o he q-K a chuk p olynomials (see [21], Vol. I I I, page 64). Acco ding o [3] (page 76) and aking
in o accoun Eq. (92) he h ee e m ecu ence ela ion o hese p olynomials can b e exp essed as
k
p
n
(
q
x
; N
) = [
q
x
(1
A
n
1
C
n
1
)]
k
p
n
1
(
q
x
; N
) +
B
n
1
k
p
n
2
(
q
x
; N
)
;
(95)
whe e
B
n
1
=
A
n
1
C
n
1
, and
A
n
=
(1 +
p q
n
)
1
q
K
+
n
(1 +
p q
2
n
) (1 +
p q
1+2
n
)
;
C
n
=
p q
1
K
+2
n
(1
q
n
)
1 +
p q
K
+
n
(1 +
p q
2
n
) ( 1 +
p q
1+2
n
)
:
The compa ison wi h (1) gi es ha
a
num
n
=
(0)
0
q
3
n
=
pq
(
pq
N
1)
q
3
n
; a
den
n
=
(0)
0
q
4
n
=
p
2
q
N
q
4
n
:
and
(
b
num
n
)
2
=
(0)
0
q
6
n
=
p
3
q
2
q
6
n
;
(
b
den
n
)
2
=
(0)
0
q
8
n
=
q
3
p
4
q
N
q
8
n
:
Then,
g
m
=
h
m
=
k
m
=
l
m
= 0 o all
m
= 0
;
1
; ::N
and
d
0
= 3
; e
0
= 4
;
0
= 6
; s
0
= 8
:
This is he case
d
0
e
0
=
1
<
0 and
0
s
0
=
2
<
0, i.e., case 2(a)i. The e o e, Eq. (34) o
Theo em 3 gi es us he alues
0
m
(1) =
8
>
<
>
:
1
m
= 0
0
m
1
(96)
o he momen s o he asymp o ic densi y o ze os
1
(
x
). Fu he mo e, since
1
=
1
2
(
0
s
0
) =
m
,
2
=
(
P
j
k
=1
k
0
k
+1
4
P
j
1
k
=1
k
k
+1
) and
M
= 0, Eq. (33) o Theo em 3 gi es us
0
m
=
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
q
2
"
q
(
pq
N
1)
pq
N
#
R
0
"
q
1
N
p
#
R
1
q
m
1
; m
0
;
(97)
o he momen s o he sp ec al quan i y
(
x
) dened by Eq. (31).
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
23
q-Racah p olynomials
R
n
(
(
x
)
; ; ; ;
)
.
(
x
) =
q
x
+
q
x
+1
.
I is well known he imp o an ole ha he 6j symb ols play in he quan um angula momen um
heo y (see [6]). I is known ha he q-analog o he Racah co ecien s (6j symb ols) o he q-
algeb a
U
q
(
sl
2
) a e p op o ional o he q-Racah p olynomials (see [21 ], Vol. I I I, page 70). F om
he h ee e m ecu ence ela ion o hese p olynomials [3] (page 53), as well as Eq. (92), we can
ew i e [3] Eq.(3.15.3) in he o m
(
x
)
R
n
1
(
(
x
)
; ; ; ;
) =
R
n
(
(
x
)
; ; ; ;
)+
+[1 +
q
(1
A
n
1
C
n
1
)]
R
n
1
(
(
x
)
; ; ; ;
) +
B
n
1
R
n
2
(
(
x
)
; ; ; ;
)
;
(98)
whe e
B
n
1
=
A
n
1
C
n
1
, and
A
n
=
1
q
1+
n
1
q
1+
n
1
q
1+
n
1
q
1+
n
(1
q
1+2
n
) (1
q
2+2
n
)
;
C
n
=
q
(1
q
n
) (
q
n
) (1
q
n
) (
q
n
)
(1
q
2
n
) ( 1
q
1+2
n
)
:
The compa ison wi h (1) gi es ha
a
num
n
=
(0)
0
q
3
n
=
q
(
+
+
+
+
+
+
+
)
q
3
n
;
a
den
n
=
(0)
0
q
4
n
=
2
2
q
4
n
:
and
(
b
num
n
)
2
=
(0)
0
q
8
n
=
q
4
4
q
8
n
;
(
b
den
n
)
2
=
(0)
0
q
8
n
=
4
4
q
8
n
:
Then,
g
m
=
h
m
=
k
m
=
l
m
= 0 o all
m
= 0
;
1
; ::N
and
d
0
= 3
; e
0
= 4
;
0
= 8
; s
0
= 8
:
This is he case
d
0
e
0
=
1
<
0 and
0
s
0
= 0, i.e., case 2(a)iii. The e o e, Eq (16) o Theo em
3 yield he momen s
0
m
(1) =
8
>
>
>
<
>
>
>
:
1
m
= 0
X
(
m
)
F
(0
;
1
;
0
; :::;
j
;
0) [
q
]
R
m
1
(99)
o he asymp o ic densi ies o ze os
1
(
x
) dened by Eq. (31).
q-Askey & Wilson p olynomials
p
n
(
x; a; b; c; d
)
.
Acco ding o [3 ] (page 51) and Eq. (92), he h ee e m ecu ence ela ion o he q-Askey &
Wilson p olynomials can b e ew i en as
xp
n
1
(
x; a; b; c; d
) =
p
n
(
x; a; b; c; d
) +
1
2
[
a
+
a
1
(
A
n
1
+
C
n
1
)]
p
n
1
(
x; a; b; c; d
)+
+
B
n
1
p
n
2
(
x; a; b; c; d
)
;
(100)
whe e
B
n
1
=
A
n
1
C
n
1
, and
A
n
=
1
a b c d q
1+
n
(1
a b q
n
) ( 1
a c q
n
) ( 1
a d q
n
)
a
(1
a b c d q
2
n
) (1
a b c d q
1+2
n
)
;
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
24
C
n
=
a
1
bcq
1+
n
1
b d q
1+
n
1
c d q
1+
n
(1
q
n
)
(1
a b c d q
2+2
n
) ( 1
a b c d q
1+2
n
)
:
The compa ison wi h (1) gi es
a
num
n
=
(0)
0
q
3
n
=
q abcd
(
abc
+
abd
+
acd
+
bcd
+
q
(
a
+
b
+
c
+
d
))
q
3
n
;
a
den
n
=
(0)
0
q
4
n
= 2
a
2
b
2
c
2
d
2
q
4
n
:
and
(
b
num
n
)
2
=
(0)
0
q
8
n
=
a
4
b
4
c
4
d
4
q
8
n
;
(
b
den
n
)
2
=
(0)
0
q
nn
=
a
4
b
4
c
4
d
4
q
8
n
:
Then,
g
m
=
h
m
=
k
m
=
l
m
= 0 o all
m
= 0
;
1
; ::N
and
d
0
= 3
; e
0
= 4
;
0
= 8
; s
0
= 8
:
This is he case
d
0
e
0
=
1
<
0 and
0
s
0
= 0, i.e., case 2(a)iii. The e o e, Eq. (38) o Theo em
3 gi es us he momen s
0
m
(1) =
8
>
>
>
<
>
>
>
:
1
m
= 0
X
(
m
)
F
(0
;
1
;
0
; :::;
j
;
0)
m
1
:
(101)
o he asymp o ic densi y o ze os
1
(
x
) dened by Eq. (31).
Al Salam and Ca li z p olynomials
u
n
(
x
)
and
n
(
x
)
.
In dealing wi h he q-ha monic oscilla o , Askey and Suslo [16 ] ha e in o duced he q-p olynomials
u
n
(
x
) =
n
q
n
(
n
1)
2
U
(
)
n
(
x
)
:
whe e
U
n
(
x
) a e he so called Al Salam and Ca li z p olynomials. These p olynomials sa is y he
ecu ence ela ion [16]
xu
n
1
(
x
) =
u
n
(
x
) + (1
)
q
n
1
u
n
1
(
x
) +
q
n
2
(1
q
n
1
)
u
n
2
(
x
)
;
(102)
which is o he yp e (1) wi h he co ecien s
a
num
n
=
(0)
0
q
n
= (1
)
q
1
q
n
; a
den
n
= 1
;
and
(
b
num
n
)
2
=
(0)
0
q
2
n
=
q
1
q
2
n
;
(
b
den
n
)
2
=
(0)
0
q
s
0
n
= 1
:
Then,
g
m
=
h
m
=
k
m
=
l
m
= 0 o all
m
= 0
;
1
; ::N
and
d
0
= 1
; e
0
= 0
;
0
= 2
; s
0
= 0
:
This is he case
d
0
e
0
= 1 and
0
s
0
= 2, i.e., case 2(b)iiB. The e o e, Eqs. (49) and (50) o
Theo em 4 gi e us he momen s
m
(1) =
8
>
>
>
>
<
>
>
>
>
:
1
m
= 0
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
) [1
]
R
0
R
q
2
q
m
1
m
1
(103)
THE DISTRIBUTION OF ZEROS OF GENERAL Q-POLYNOMIALS
25
and
++
m
(1) =
8
>
>
>
<
>
>
>
:
1
m
= 0
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
) [1
]
R
0
R
q
2
m
1
(104)
(whe e
2
=
P
j
k
=1
k
0
k
+1
+ 4
P
j
1
k
=1
k
k
+1
m
) co esp onding o he asymp o ic quan i ies
1
(
x
)
and
++
1
(
x
), esp ec i ely.
I has b een encoun e ed [15 ] ha ano he class o Al Salam and Ca li z p olinomials, o b e de-
no ed by
n
(
x
), is ela ed also o he q-oscilla o . So, i seems na u al o sea ch o i s dis ibu ion
o ze os. These p olynomials sa is y he ela ion [15 ]
x
n
1
(
x
) =
n
(
x
) + (
q
+
)
q
n
2
n
1
(
x
) +
q
n
3
(
q
n
1
1)
n
2
(
x
)
:
(105)
The e o e,
d
0
=
1
; e
0
= 0
;
0
=
1
; s
0
= 0. This co esp onds o he case 2(a)i. Then, Eq. (34)
o Theo em 3 gi es us he momen s
0
m
(1) =
8
>
<
>
:
1
m
= 0
0
m
1
:
(106)
o he asymp o ic densi y
(
x
). Fu he mo e, since
1
=
1
2
(
R
0
+
m
),
2
=
(
P
j
k
=1
k
0
k
+1
2
P
j
1
k
=1
k
k
+1
) and
M
= 0, Eq. (33) o Theo em 3 gi es
0
m
=
X
(
m
)
F
(
0
1
;
1
; :::;
0
j
+1
)
q
2
h
q
1
(
q
+
)
i
R
0
R
q
m
q
1
2
(
R
0
+
m
)
1
(107)
o he momen s o he no malized - o-
1
N
sp ec al quan i y
1
(
x
) dened by Eq. (31).
The li le q-Jacobi p olynomials
p
n
(
x; a; b
)
.
The li le q-Jacobi p olynomials
p
n
(
x; a; b
) play a undamen al ole (see e.g. [20]) in he Rep esen-
a ion Theo y o he q-Algeb a
U
q
(
sl
2
) b ecause hey a e he ma ix elemen s o he ep esen a ions
T
l
(see [21 ], Vol. I I I, page 51). They sa is y he h ee e m ecu ence ela ion [3] (page 59)
p
n
(
x; a; b
) = [
x
+
A
n
1
+
C
n
1
]
p
n
1
(
x; a; b
) +
B
n
1
p
n
2
(
x; a; b
)
;
(108)
whe e A and C pa ame e s a e gi en by
A
n
=
q
n
1
a q
1+
n
1
a b q
1+
n
(1
a b q
1+2
n
) (1
a b q
2+2
n
)
;
C
n
=
a q
n
(1
q
n
) ( 1
b q
n
)
(1
a b q
2
n
) ( 1
a b q
1+2
n
)
;
and
B
n
=
A
n
1
C
n
. This ela ion is o he yp e (1) wi h he co ecien s
a
num
n
=
(0)
0
q
3
n
=
ab
(1 +
a
)
q
3
n
; a
den
n
=
(0)
0
q
4
n
=
a
4
b
4
q
4
n
:
and
(
b
num
n
)
2
=
(0)
0
q
6
n
=
a
3
b
2
q
6
n
;
(
b
den
n
)
2
=
(0)
0
q
8
n
=
q a
4
b
4
q
8
n
: