q

Classial p olynomials and he
q

Askey and
Niki o o -U a o Tableaus
R.

Al a ez-No da se
a;b
1
and J. C. Medem
a
2
a
Depa amen o de Analisis Ma ema io. Uni e sidad de Se illa.
Ap do. 1160, E-41080 Se illa, Spain
b
Ins i u o Ca los I de Fsia Teo ia y Compu aional, Uni e sidad de G anada,
E-18071 G anada, Spain
Janua y 8, 2000
Abs a
In his pap e we on inue he s udy o he
q

lassial (dis e e) p olynomials (in he Hahn's
sense) s a ed in [18℄. He e we will ompa e ou sheme wi h he well known
q

Askey Sheme
and he Niki o o -U a o Tableau. Also, new amilies o
q

p olynomials a e in o dued.
In o du ion
The so-alled
q

p olynomials ons i u e a e y imp o an and in e es ing se o sp eial un ions
and mo e sp eially o o hogonal p olynomials. They app ea in se e al b anhes o he na u al
sienes, e.g., on inued a ions, Eule ian se ies, he a un ions, ellip i un ions,...; see [3 , 9℄,
quan um g oups and algeb as [14, 15, 25 ℄, among o he s (see also [10 , 20 ℄). They ha e b een
in ensi ely s udied in he las yea s by se e al p eople (see e.g. [13℄) using se e al o ols. One o
hem is he one e iewed in [13 ℄ whih is based on he basi hyp e geome i se ies [10 ℄ and was
de elop ed mainly by he Ame ian Sho ol s a ing by he wo ks o And ews and Askey (see e.g.
[4℄, he li e a u e on his me ho d is so as ha we a e no able o inlude i he e, a e y omple e
lis is gi en in [13℄) and lead o he so-alled
q

Askey Tableau o hyp e geome i p olynomials
[13℄. In o he di e ion, he Russian ( o me So ie ) sho ol, s a ing om he wo ks by Niki o o
and U a o [21 ℄ and u he de elop ed by A akishiye and Suslo (see e.g. [5 , 6, 20, 23 , 24℄ and
e e enes on ained he ein), ha e onside ed he die ene analog in non-uni o m la ies o he
hyp e geome i die en ial equa ion [22℄, om whe e he hyp e geome i ep esen a ion o he
q

p olynomials ollows in a e y simple way [5, 23℄. This shema leads o he Niki o o -U a o
ableau [20 , 23℄ o he p olynomial solu ions o he die ene hyp e geome i equa ion on non-
uni o m la ies. A sp eial men ion dese es he pap e by A akishie and Suslo [6 ℄ whe e a
die ene analog o he well known me ho d o unde e mina ed o eÆien s ha e b een de elop ed
o he hyp e geome i equa ion on non-uni o m la ies and also gi e a lassia ion simila o
he Niki o o and U a o 1991 one bu o he
q

sp eial un ions (no only o he p olynomials
solu ions).
Ou main aims he e a e wo: o on inue he s udy s a ed in [18℄ using he algeb ai heo y
de elop ed by Ma oni [16℄ and o lassi y he
q

lassial p olynomials and ompa e wi h he
q

Askey
and Niki o o & U a o Tableaus. In a , in [18 ℄ we ha e p o en se e al ha a e iza ion o
he
q

lassial p olynomials as well as a e y simple ompu a ional algo i hm o nding hei
main ha a e is is (e.g. he o eÆien s o he h ee- e m eu en ela ion, s u u e ela ion
o Al-Salam Chiha a, e ). Going u he , we will gi e he e a e y na u al" lassia ion o he
q

lassial p olynomials in o dued by Hahn in his pap e [11℄, i.e., we will lassi y all o hogonal
p olynomial sequenes suh ha hei
q

die enes, dened by 
(
x
) =
(
q x
)

(
x
)
(
q

1)
x
a e o hogonal
in he widesp ead sene: he
q

Hahn Tableau (a  s s ep on his in he ame wo k o he
q

Askey ableau was done in [15℄). No ie ha he a o esaid p olynomials a e ins anes o he
1
Phone: +34 954 55 7997, Fax: 954 55 7972. E-mail: ania.es
2
Phone: +34 954 55 7997, Fax: 954 55 7972. E-mail: jmedemia.es
1
2
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
q

p olynomials on he linea exp onen ial la ie
x
(
s
) =

1
q
s
. Fo se e al su eys on his la ie and
hei o esp onding p olynomials see [2, 4, 5 , 8 , 10 , 13, 20, 24℄. (see also se ion 3.2 om b elow).
Fu he mo e, we will ompa e ou lassia ion (
q

Hahn Tableau) wi h he a o esaid wo Shemas.
F om his ompa a ion we nd ha he e a e missing amilies in he
q

Askey Shema (one o hem
is a non-p osi i e deni e amily) and using he esul s o [18 ℄ we s udy hem wi h de ails. Also he
o esp ondene o his
q

Hahn Shema and he Niki o o & U a o one will b e s ablished. In suh
a way a omple e o esp ondene b e ween he
q

lassial amilies o he
q

Askey and Niki o o
& U a o Tableaus o exp onen ial linea la ies will b e shown.
The s u u e o he pap e is as ollows. In Se ion 1 we in o due some no a ions and deni ions
use ul o he nex ones. In Se ion 2, he
q

weigh un ions a e in o dued and ompu ed
o all
q

lassial amilies. This will allow o lassi y all o hogonal p olynomial amilies o he
q

Hahn ableau. Finally, in Se ion 3, se e al applia ions a e onside ed: he lassia ion o he
q

lassial p olynomials (
q

Hahn Tableau), he in eg al ep esen a ion o he o hogonali y, he
hyp e geome i ep esen a ion o hese
q

lassial p olynomials as well as he de ailed s udy o wo
new amilies o
q

p olynomials.
1 P elimina ies
In his se ion we will gi e a b ie su ey o he op e a ional alulus and some basi onep s
and esul s needed o he es o he wo k.
Le
P
b e he linea spae o p olynomial un ions in
C
wi h omplex o eÆien s and
P

b e
i s algeb ai dual spae, i.e.,
P

is he linea spae o all linea applia ions
u
:
P
!
C
. In he
ollowing we will e e o he elemen s o
P

as un ionals and we will deno e hem wi h b old le e s
(
u
;
;:::
).
Sine he elemen s o
P

a e linea un ionals, i is p ossible o de e mine hem om hei
a ions on a gi en basis (
B
n
)
n

0
o
P
, e.g. he anonial basis o
P
, (
x
n
)
n

0
. In gene al, we will
ep esen he a ion o a un ional o e a p olynomial by o mula
h
u
; 
i
;
u
2
P

; 
2
P
, and
he e o e a un ional is omple ely de e mined by a sequene o omplex numb e s
h
u
; x
n
i
=
u
n
,
n

0, he so-alled momen s o he un ional.
Deni ion 1.1
Le
(
P
n
)
n

0
be a basis sequene o
P
. We say ha
(
P
n
)
n

0
is an o hogonal
polynomial sequene (OPS in sho ), i and only i he e exis s a un ional
u
2
P

suh ha
h
u
; P
m
P
n
i
=
k
n
Æ
mn
,
k
n
6
= 0
; n

0
, whe e
Æ
mn
is he K oneke del a. I
k
n
>
0
o al l
n

0
,
we say ha
(
P
n
)
n

0
is a posi i e deni e OPS.
Deni ion 1.2
Le
u
2
P

be a un ional. We say ha
u
is a quasi-deni e un ional i and only
i he e exis s a polynomial sequene
(
P
n
)
n

0
, whih is o hogonal wi h espe o
u
. I
(
P
n
)
n

0
is
posi i e deni e, we say ha
u
is a posi i e deni e un ional.
Deni ion 1.3
Gi en a polynomial sequene
(
P
n
)
n

0
, we say ha
(
P
n
)
n

0
is a moni o hogonal
polynomial sequene (MOPS in sho ) wi h espe o
u
, and we deno e i by
(
P
n
)
n

0
= mops
u
i
and only i
P
n
(
x
) =
x
n
+ lowe deg ee e ms
and
h
u
; P
m
P
n
i
=
k
n
Æ
nm
; k
n
6
= 0
; n

0
.
Also he nex heo em will b e use ul
Theo em 1.1
(
Fa a d Theo em [7℄)
Le
(
P
n
)
n

0
be a moni polynomial basis sequene. Then,
(
P
n
)
n

0
is an MOPS i and only i he e exis wo sequenes o omplex numbe s
(
d
n
)
n

0
and
(
g
n
)
n

1
, suh ha
g
n
6
= 0
,
n

1
and
xP
n
=
P
n
+1
+
d
n
P
n
+
g
n
P
n

1
; P

1
= 0
; P
0
= 1
; n

0
;
(1.1)
whe e
P

1
(
x
)

0
and
P
0
(
x
)

1
. Mo eo e , he un ional
u
wi h espe o whih he polynomials
(
P
n
)
n

0
a e o hogonal is posi i e deni e i and only i
(
d
n
)
n

0
is a eal sequene and
g
n
>
0
o
al l
n

1
.
R.

Al a ez-Noda se and J. C. Medem
3
In he ollowing, we will use he no a ion:
Deni ion 1.4
Le

2
P
and
a
2
C
,
a
6
= 0
. We al l he ope a o
H
a
:
P
!
P
,
H
a

(
x
) =

(
ax
)
,
a dila ion o a io
a
2
C
n
0
g
.
This op e a o is linea on
P
and sa ises H
a
(
 
) = H
a


H
a

. Also no ie ha o any omplex
numb e
a
6
= 0, H
a

H
a

1
= I, whe e I is he iden i y op e a o on
P
, i.e., o all
a
6
= 0, H
a
has an
in e se op e a o . In he ollowing we will omi any e e ene o
q
in he op e a o s H
q
and hei
in e se H
q

1
. So, H := H
q
, H

1
:= H
q

1
.
Nex , we will dene he so alled
q

de i a i e op e a o [11 ℄. We will supp ose also ha
j
q
j 6
= 1
(al hough i is p ossible o weak his ondi ion).
Deni ion 1.5
Le

2
P
and
q
2
C
n
0
g
,
j
q
j 6
= 1
. The
q

de i a i e ope a o

, is he ope a o
 :
P
!
P
, dened by


=
H



H
x

x
=
H



(
q

1)
x
:
The
q

1

de i a i e ope a o

?
, is he ope a o

?
:
P
!
P
dened by

?

=
H

1



H

1
x

x
=
H

1



(
q

1

1)
x
:
In his way,


and

?

wil l deno e he
q

de i a i e and
q

1

de i a i e o

, espe i ely.
The ab o e wo op e a o s  and 
?
a e linea op e a o s on
P
, and

x
n
=
H
x
n

x
n
(
q

1)
x
=
(
q
n

1)
x
n
(
q

1)
x
= [
n
℄
x
n

1
; n >
0
;
1 = 0
;
(1.2)
i.e., 

2
P
. He e [
n
℄
; n
2
N
, deno es he basi
q

numb e
n
dened by
[
n
℄ =
q
n

1
q

1
= 1 +
q
+
:::
+
q
n

1
; n >
0
;
[0℄ = 0
:
(1.3)
Also he
q

1
numb e s [
n
℄
?
, dened by [
n
℄
?
=
q

n

1
q

1

1
=
q
1

n
[
n
℄ will b e used.
No ie ha 
?
is no he in e se o . In a hey a e ela ed by H 
?
= 
;
H

1
=
?
.
The
q

de i a i e sa ises he p o du ule (
 
) =



+ H




= H




+



.
Deni ion 1.6
Le
!
a de i able un ion a
x
= 0
suh ha
8
a
2
dom
!
,
aq
2
dom
!
. Then, we
wil l dene he
q

de i a i e o
!
by he exp ession

!
=
H
!

!
H
x

x
=
H
!

!
(
q

1)
x
; x
6
= 0
;

!
(0) =
!
0
(0)
:
(1.4)
Deni ion 1.7
Le
u
2
P

and

2
P
. We dene he a ion o a dila ion
H
a
and he
q

de i a i e

on
P

by he exp essions
H
a
:
P

!
P

,
h
H
a
u
; 
i
=
h
u
;
H
a

i
,
 :
P

!
P

,
h

u
; 
i
=
h
u
;


i
,
espe i ely.
Deni ion 1.8
Le
u
2
P

and

2
P
. We dene a polynomial modia ion o a un ional
u
, he
un ional

u
,
h

u
; 
i
=
h
u
;  
i
;
8

2
P
.
No ie ha we use he same no a ion o he op e a o s on
P
and
P

. Whene e i is no sp eied
on whih linea spae an op e a o a s, i will b e unde s o o d ha i a s on he p olynomial spae
P
.
Deni ion 1.9
Le
u
2
P

be a quasi-deni e un ional and
(
P
n
)
n

0
= mops(
u
)
. We say ha
u
o
(
P
n
)
n

0
a e
q

lassi un ional o MOPS, espe i ely, i and only i he sequene
(
P
n
+1
)
n

0
is also o hogonal.
4
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
No ie ha in he Hahn deni ion [11℄
q
is a eal pa ame e and he e, in gene al,
q
2
C
n
0
g
,
j
q
j 6
= 1.
In he ollowing (
Q
n
)
n

0
will deno e he sequene o moni
q

de i a i es o (
P
n
)
n

0
, i.e.,
Q
n
=
1
[
n
+1℄
P
n
+1
, o all
n

0.
Theo em 1.2
(Medem e al. [17, 18 ℄) Le
u
2
P

be a quasi-deni e un ional. and
(
P
n
)
n

0
=
mops(
u
)
. Then, he ol lowing s a emen s a e equi alen :
(a)
u
and
(
P
n
)
n

0
a e, espe i ely, a
q

lassial un ional and a
q

lassial MOPS.
(b) The e exis s a pai o polynomials

and
,
deg


2
,
deg
= 1
, suh ha
(

u
) =
u
:
(1.5)
()
(
P
n
)
n

0
sa ises he
q
 SL
die ene equa ion


?
P
n
+

?
P
n
=
b

n
P
n
; n

0
;
(1.6)
i.e.,
P
n
a e he eigen un ions o he S u m-Liou il le ope a o


?
+

?
o esponding o
he eigen alues
b

n
.
Mo eo e , i

(
x
) =
b
ax
2
+ 
ax
+ _
a;
(
x
) =
b
bx
+

b;
b
b
6
= 0
;
(1.7)
hen, he quasi-deni eness o
u
implies
[
n
℄
b
a
+
b
b
6
= 0
and he ol lowing equi alenes hold
[
n
℄
b
a
+
b
b
6
= 0
; n

0
()
b

n
6
=
b

m
;
8
n; m

1
; n
6
=
m
()
b

n
6
= 0
;
8
n

1
:
Theo em 1.3
Le
u
2
P

, be a quasi-deni e un ional,
(
P
n
)
n

0
= mops
u
and
Q
(
k
)
n
=
1
[
n
+1℄
(
k
)

k
P
n
+
k
,
whe e
[
n
+ 1℄
(
k
)

[
n
+ 1℄[
n
+ 2℄
:::
[
n
+
k

1℄
. The ol lowing s a emen s a e equi alen :
(a)
(
P
n
)
n

0
is
q

lassial, (b)
(
Q
(
k
)
n
)
n

0
is
q

lassial,
k

1
.
Mo eo e , i
u
sa ises he equa ion
(

u
) =
u
,
deg


2
and
deg
= 1
, hen
(
Q
(
k
)
n
)
is
o hogonal wi h espe o
(
k
)
= H
(
k
)


u
,
H
(
k
)
=
Q
k
i
=1
H
i

1

, and i sa ises
(

(
k
)
(
k
)
) =
(
k
)
(
k
)
;
deg

(
k
)

2 deg
(
k
)
= 1
;
whe e

(
k
)
= H
k

and
(
k
)
=
+ 
P
k

1
i
=0
H
i

, and hey a e he polynomial solu ions o he
q
 SL
equa ion
SL
(
k
)
Q
(
k
)
n
=

(
k
)

?
Q
(
k
)
n
+
(
k
)

?
Q
(
k
)
n
=
b

(
k
)
n
Q
(
k
)
n
;
(1.8)
whe e he polynomials

(
k
)
and
(
k
)
and he eigen alues
b

(
k
)
n
a e

(
k
)
=
q
2
k
b
ax
2
+
q
k
ax
+

a
;
(
k
)
= ([2
k
℄
b
a
+
b
b
)
x
+ ([
k
℄
a
+
b
)
;
b

(
k
)
n
= [
n
℄
?
([2
k
+
n

1℄
b
a
+
b
b
)
:
(1.9)
Fu he mo e, in [17 , 18℄ he ollowing esul was p o en:
Theo em 1.4
Le
u
2
P

, be a quasi-deni e un ional,
(
P
n
)
n

0
= mops
u
,
; 
?
;
2
P
, suh
ha

?
=
q

1

+ (
q

1

1)
x
,
deg


2
,
deg

?

2
and
deg
= 1
. Then, he ol lowing s a emen s
a e equi alen
(a)
u
and
(
P
n
)
n

0
= mops
u
a e
q

lassial and
(

u
) =
u
,
(b)
u
and
(
P
n
)
n

0
= mops
u
a e
q

1

lassial and

?
(

?
u
) =
u
.
R.

Al a ez-Noda se and J. C. Medem
5
() The e exis a polynomial

2
P
,
deg


2
and h ee sequenes o omplex numbe s
a
n
; b
n
; 
n
,

n
6
= 0
, suh ha


P
n
=
a
n
P
n
+1
+
b
n
P
n
+

n
P
n

1
; n

1 ; (1.10)
(d) he e exis a omplex numbe s
e
n
; h
n
, suh ha
P
n
=
Q
n
+
e
n
Q
n

1
+
h
n
Q
n

2
; n

2
:
(1.11)
(e) The e exis a polynomial

2
P
,
deg


2
and a sequene o omplex numbe s
n
,
n
6
= 0
,
n

1
suh ha
P
n
u
=
n

n
(H
(
n
)


u
)
;
H
(
n
)

=
n
Y
i
=1
H
i

1
 ;
n
=
q
(
n
2
)
n
Y
i
=1

[2
n

i

1℄
b
a
+
b
b


1
; n

1
:
(1.12)
2 The
q

weigh un ion
!
2.1 Deni ion and  s p op e ies
In his se ion we will onside he so-alled weigh un ions o
q

lassial p olynomials. The nex
p op osi ion an b e p o en s aigh o wa d (see e.g. [12 ℄).
P op osi ion 2.1
Le
!
a un ion suh ha i
a
2
dom
!
,
aq

1
2
dom
!
and ha sa ises he
die ene equa ion

?
(
!
) =
q !
()
!
=
q
H(

?
!
)
; ;
2
P
; 
?
=
q

1

+ (
q

1

1)
x :
(2.1)
Then, he ol lowing wo equa ions a e equi alen


?
P
n
+

?
P
n
=
b

n
P
n
;
()

?
(
!

P
n
) =
q
b

n
! P
n
; n

1
:
(2.2)
The ab o e p op osi ion allows us o gene alize he lassial p o edu e o he
q

ase o ob aining
almos all he ha a e is is o he MOPS. The equa ion (2.1) is usually alled he
q

Pea son
equa ion and i s solu ion
!
is known as he
q

weigh un ion and i allows o ew i e he S u m-
Liuo ille equa ion (1.6) in i s sel -adjoin o m (2.2). Mo eo e , he weigh un ion
!
allow us o
ob ain he s anda d"
q

Ro d igues o mula and also jus i y he
q

in eg al ep esen a ion o he
o hogonali y ela ion. In suh a way i is na u al o gi e he ollowing
Deni ion 2.1
Le
u
2
P

, be a quasi-deni e un ional sa is ying he dis ibu ional equa ion
(1.5), whe e
;
2
P
,
deg


2
,
deg
= 1
and
(
P
n
)
n

0
= mops
u
. We say ha
!
is he
q

weigh
un ion assoia ed o
u
( espe i ely o
(
P
n
)
n

0
) i
!
sa ises he equa ion (2.1)

?
(
!
) =
q !
.
The las deni ion allows us o ew i e he
q
 SL
equa ion (1.8) in i s sel -adjoin o m. In
a , an s aigh o wa d alula ions show ha , i
!
(
k
)
sa ises he
q

Pea son equa ion

?
(

(
k
)
!
(
k
)
) =
q
(
k
)
!
(
k
)
;
(2.3)
whe e

(
k
)
and
(
k
)
a e gi en in (1.9), hen (1.8) an b e ew i en in i s sel -adjoin o m

?
(

(
k
)
!
(
k
)

Q
(
k
)
n
) =
q
b

(
k
)
n
!
(
k
)
Q
(
k
)
n
; n

1
; k
= 0
;
1
;::: ;n:
(2.4)
P op osi ion 2.2
Le
!
he solu ion o (2.1) and
!
(
k
)
he solu ion o (2.3). Then,
!
(
k
)
=

(
n

1)
!
(
n

1)
=

= H
(
n
)


! ; !
(0)

! :
(2.5)

6
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
P o o :
We s a om he
q

Pea son equa ion (2.3) and ew i e i in i s equi alen o m

(
k
)
!
(
k
)
=
q
H[

(
k
)
℄
?
H
!
(
k
)
, whe e [

(
k
)
℄
?
=
q

1

(
k
)
+ (
q

1

1)
x
(
k
)
=

?
, o all
k
2
N
. Thus, by subs i u ing
!
(
k
)
= H
(
n
)


!
in

(
k
)
!
(
k
)
=
q
H

?
H
!
(
k
)
, we nd

(
k
)
!
(
k
)
=
q
H[

(
k
)
℄
?
H
!
(
k
)
()
H
k

(

H


H
k

1


!
) =
q
H

?
H


H
k

H
!
()
!
=
q
H

?
H
!
()

?
(
!
) =
q ! ;
om whe e he p op osi ion ollows.

Rema k 2.1
No ie ha he polynomials
(

(
k
)
)
?
and
(

?
)
(
k
)
a e e y die en . In a , he  s
one oge he wi h
(
k
)
a e he o esponding polynomials ha appea in he
q

1

dis ibu ional
equa ion sa ised by he un ional
(
k
)
, i.e., he un ional wi h espe o whih he
k

h moni
de i a i es
Q
(
k
)
n
a e o hogonal, (see P oposi ion 1.4)
(

(
k
)
(
k
)
) =
(
k
)
(
k
)
()

?
(

(
k
)
)
?
(
k
)
=
(
k
)
(
k
)
;
(

(
k
)
)
?
=

?
;
8
k
2
N
;
whe eas he seond one join wi h
(
?
)
(
k
)
a e he polynomial oeÆien s o he
q

1
 SL
equa ion
(
?
)
(
k
)

?

Q
?
(
k
)
n
(
?
)
(
k
)
= (
b

?
)
(
k
)
n
Q
?
(
k
)
n
, o he
n

h
q

1

de i a i e
Q
?
(
k
)
n
o he polynomials
P
n
,
Q
?
(
k
)
n
=
1
[
n
+1℄
?
(
k
)
[
?
℄
n
P
n
+
k
o he
q

1

dis ibu ional equa ion sa ised by he un ional
?
(
k
)
,

?
[(

?
)
(
k
)
?
(
k
)
℄ = (
?
)
(
k
)
?
(
k
)
;
(

?
)
(
k
)
= H

k

?
;
8
k
2
N
;
i.e., he un ional wi h espe o whih he
k

h moni de i a i es
Q
?
(
k
)
n
a e o hogonal.
2.2 Compu a ion o he
q

weigh un ions
This se ion is de o ed o ob ain he
q

weigh un ion asso ia ed o all
q

lassial un ionals,
i.e., he quasi-deni e un ionals o esp onding o he MOPS in he widesp ead sense

u
; P
2
n

6
= 0,
o all
n

0. In a , Theo em 2.1 and 2.2 will gi e, in a e y na u al way, he key o he
lassia ion o all
q

lassial o hogonal p olynomials.
In he ollowing we onside he ase when
j
q
j
<
1 (
j
q

1
j
>
1). Also we will use he s anda d
no a ion (
a
;
q
)
n
= (1

a
)(1

aq
)

(1

aq
n

1
) o
n

1, (
a
;
q
)
0

1 o he
q

analogue o he
Po hamme symb ol, and (
a
;
q
)
1
=
Q
1
n
=0
(1

aq
n
), o he absolu ely on e gen inni e p o du
o
j
q
j
<
1.
Fi s o all, we will ew i e he
q

Pea son equa ion (2.1)

?
(
!
) =
q !
()
!
=
q
H

?
H
!
()

?
!
=
q

1
H

1

H

1
! ;
(2.6)
and sol e he esul ing equa ion by he eu en p o edu e shown in gu e 1.
Figu e 1.
Reu en shema using he
q

dila ion.
w
= H
n
w

q
H

?


H
q
H

?


:::

H
n

1
q
H

?

|{z }
H
(
n
)
q
H

?


=
Q
n

1
k
=0
q 
?
(
q
k
+1
x
)

(
q
k
x
)

H
2

H
2
w
= H
2
(
q
H

?
)H
3
w
::: :::












1





R









1


R
H

H
w
= H(
q
H

?
)H
2
w
?
H
w
= H
2
w
q
H

?

H
q
H

?

w
=
q
H

?
H
w
-
?
H
w
= H
w
q
H

?

R.

Al a ez-Noda se and J. C. Medem
7
In he ase when
!
is on inuous a 0 and
!
(0)
6
= 0, aking he limi
n
! 1
, we nd, sine
lim
n
!1
H
n
w
= lim
n
!1
w
(
q
n
x
) =
w
(0),
!
=
!
(0) lim H
(
1
)
q
H

?

=
!
(0) lim
n
!1
H
(
n
)
q
H

?

=
!
(0)
1
Y
n
=0
q
H

?

:
(2.7)
The nex s ep is o ob ain an explii exp ession o he p o du H
(
1
)
q
H

?

. Fo doing ha we need
a lemma whih is in e es ing in i s own igh .
Lemma 2.1
I

is an
n

h deg ee polynomial wi h an independen e m

(0) = 1
, and ze os
a
i
2
C
n
0
g
,
i
= 1
;
2
;::: ;n
, hen
H
(
1
)

= (
a

1
1
x
;
q
)
1
(
a

1
2
x
;
q
)
1

(
a

1
n
x
;
q
)
1
:= (
a

1
1
x; a

1
2
x;

; a

1
n
x
;
q
)
1
;
is an en i e un ion o
x
wi h ze os a
a
i
q

k
,
i
= 1
;
2
;::: ;n
and
k

0
. Fu he mo e, i
 =
is
a a ional un ion suh ha

(0) =

(0)
6
= 0
and wi h non- anishing ze os o i s nume a o and
denomina o , hen,
H
(
1
)


=
(
a

1
1
x
;
q
)
1
(
a

1
2
x
;
q
)
1

(
a

1
n
x
;
q
)
1
(
b

1
1
x
;
q
)
1
(
b

1
2
x
;
q
)
1

(
b

1
m
x
;
q
)
1
=
(
a

1
1
x; a

1
2
x;

; a

1
n
x
;
q
)
1
(
b

1
1
x; b

1
2
x;

; b

1
m
x
;
q
)
1
;
i is a me omo phi un ion wi h ze os a
a
i
q

k
,
i
= 1
;
2
;::: ;n
and
k

0
and poles a
b
j
q

l
,
j
= 1
;
2
;::: ;m
and
l

0
, whe e
a
i
2
C
,
i
= 1
;
2
;::: ;n
and
b
k
2
C
,
k
= 1
;
2
;::: ;m
, a e he ze os
o he nume a o and denomina o o
 =
, espe i ely.
P o o :
The p o o is based on he a ha , i

is a p olynomial o deg ee
n
wi h non anishing
ze os and

(0) = 1, hen i admi s he a o iza ion

=
A
(
x

a
1
)(
x

a
2
)

(
x

a
n
) = (

1)
n
Aa
1
a
2

a
n
|{z }

(0)=1
(1

a

1
1
x
)(1

a

1
2
x
)

(1

a

1
n
x
)
:
Then, H
(
k
)

= (
a

1
1
x; a

1
2
x;

; a

1
n
x
;
q
)
k
and so, H
(
1
)

= (
a

1
1
x; a

1
2
x;

; a

1
n
x
;
q
)
1
. This un-
ion is an en i e un ion due o he Weie s ass Theo em (see e.g. [1,
x
4.3℄). The p o o o he
seond s a emen is analogous and he un ion H
(
1
)


is me omo phi b eause is a quo ien o wo
en i e un ions (see e.g. [1,
x
4.3℄).

Now, i

(0)
6
= 0, he ab o e lemma leads us o he ollowing well known esul [11℄
Theo em 2.1
Le
(
P
n
)
n

0
= mops
u
sa is ying he
q

S u m-Liou il le equa ion (1.6). I we deno e
by
a
1
and
a
2
he ze os o

and by
a
?
1
and
a
?
2
he ze os o

?
(see P oposi ion 1.4), and al l hey a e
die en om 0, hen he ol lowing exp essions o he
q

weigh un ions
!
hold
 
?
q

weigh un ion
!
(
x
)
b
a
?
(
x

a
?
1
)(
x

a
?
2
)
,
b
a
?
a
?
1
a
?
2
6
= 0
!
(
x
) =
(
a
?
1

1
q x; a
?
2

1
q x
;
q
)
1
(
a

1
1
x; a

1
2
x
;
q
)
1
b
a
(
x

a
1
)(
x

a
2
)
,
b
aa
1
a
2
6
= 0 
a
?
(
x

a
?
1
)
,

a
?
a
?
1
6
= 0
!
(
x
) =
(
a
?
1

1
q x
;
q
)
1
(
a

1
1
x; a

1
2
x
;
q
)
1
_
a
?
6
= 0
!
(
x
) =
1
(
a

1
1
x; a

1
2
x
;
q
)
1

a
(
x

a
1
)
,

aa
1
6
= 0
!
(
x
) =
(
a
?
1

1
q x; a
?
2

1
q x
;
q
)
1
(
a

1
1
x
;
q
)
1
b
a
(
x

a
1
)(
x

a
2
)
,
b
aa
1
a
2
6
= 0
_
a
6
= 0
!
(
x
) = (
a
?
1

1
q x; a
?
2

1
q x
;
q
)
1
8
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
P o o :
Sine

(
x
) =
b
a
(
x

a
1
)(
x

a
2
) and

?
=
q

1

+ (
q

1

1)
x
=
b
a
?
(
x

a
?
1
)(
x

a
?
2
), we
ha e (
q
H

?
)(0) =
q 
?
(0) =

(0), so he p olynomials
q
H

?
and

ha e he same indep enden e m.
Using he p owe expansion o he p olynomials

and

?

(
x
) =
b
ax
2
+ 
ax
+ _
a; 
?
(
x
) =
b
a
?
x
2
+ 
a
?
x
+ _
a
?
;
we ha e
b
a
?
=
q

1
b
a
+ (
q

1

1)
b
b
, 
a
?
=
q

1

a
+ (
q

1

1)

b
and _
a
?
=
q

1
_
a
, whe e,
b
b;

b
a e he o eÆien
o he p owe expansion o
(see Eq. (1.7)). Thus,
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
deg
 <
2 =
)
b
a
= 0 =
)
b
a
?
6
= 0 =
)
deg

?
= 2
;
deg

= 2 =
)
b
a
6
= 0
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
b
b
6
=

b
a
1

q
=
)
b
a
?
6
= 0 =
)
deg

?
= 2
;
b
b
=

b
a
1

q
=
)
b
a
?
= 0
8
>
>
>
<
>
>
>
:

b
6
=


a
1

q
=
)
deg

?
= 1
;

b
=


a
1

q
=
)
deg

?
= 1
:
In all ases we an apply di e ly he ab o e lemma whih immedia ely leads us o he desi ed
esul . No ie also ha all he ob ained un ions a e me omo phi and so, hey a e on inuous
and non- anishing a
x
= 0, so we an supp ose wi hou any loss o gene ali y ha
!
(0) = 1.

In he ase when

(0) = 0, i is easy o see ha

?
(0) = 0. This ase equi es a mo e de ail
s udy. In he ollowing we should keep in mind ha o he quasi-deni eness o
u

6
0 and

and
should b e op ime p olynomials (see [18 ℄).
P op osi ion 2.3
Le
u
be a
q

lassial un ional sa is ying he dis ibu ional equa ion (1.5) wi h

=
b
ax
2
+ 
ax
,
j
b
a
j
+
j

a
j
>
0
, and
=
b
bx
+

b
,
b
b
6
= 0
. Then he ol lowing ases, ompa ible wi h he
quasi-deni eness o
u
, appea :
(a) I

=
b
ax
2
,
b
a
6
= 0
, hen,
deg

?
= 2
and i s wo ze os a e die en , o
deg

?
= 1
.
(b) I

=
b
ax
2
+ 
ax
,
b
a

a
6
= 0
, hen,
deg

?
= 2
, o
deg

?
= 1
.
() I

= 
ax
,

a
6
= 0
, hen,
deg

?
= 2
.
P o o :
(a) Sine

=
b
ax
2
, hen
=
b
bx
+

b
, wi h

b
6
= 0, o he wise
di ides

. The e o e,

?
= (
q

1
b
a
+
(
q

1

1)
b
b
)
x
2
+ (
q

1

1)

bx
has a non- anishing o eÆien on
x
. I
b
b
6
=

b
a
1

q
hen
b
a
?
6
= 0 and
deg

?
= 2 and

?
has wo die en ze os one o whih is lo a ed a he o igin. I
b
b
=

b
a
1

q
hen
deg

?
= 1.
The o he wo ases a e p o en analogously.

The nex s ep is o nd he
q

weigh un ions o all p ossible ases ao ding wi h he ab o e
p op osi ion ( ememb e ha

(0) = 0 =

?
(0)). The e a e wo la ge lasses. Class I o esp onding
o he ase when

and

?
ha e non- anishing e m on
x
and I I when hey ha e a anishing e m
on
x
.
I.
We s a wi h he ase when

and

?
ha e no - anishing e m on
x
. In his ase he e a e h ee
die en p ossibili ies (sub lasses):
(a)

(
x
) =
b
ax
(
x

a
1
),
b
aa
1
6
= 0 and

?
(
x
) =
b
a
?
x
(
x

a
?
1
),
b
a
?
a
?
1
6
= 0,
(b)

(
x
) =
b
ax
(
x

a
1
),
b
aa
1
6
= 0 and

?
(
x
) = 
a
?
x
, 
a
?
6
= 0,
()

(
x
) = 
ax
, 
a
6
= 0 and

?
(
x
) =
b
a
?
x
(
x

a
?
1
),
b
a
?
a
?
1
6
= 0.
R.

Al a ez-Noda se and J. C. Medem
9
To nd he o esp onding
q

weigh un ions we will ew i e he quo ien
q
H

?
=
=
xq
(H

?
)
0
=x
0
, whe e

=
x
0
and H

?
=
x
(H

?
)
0
. In gene al,

0
(0)
6
= (H

?
)
0
(0), so, in o de
o apply a me ho d, simila o he one used o p o e Theo em 2.1, we will assume ha
!
an b e
ew i en on he o m
!
=
j
x
j

!
0
,

2
C
n
0
g
, whe e

is a ee pa ame e o b e ound. An
s aigh o wa d alula ions show ha i
!
sa ises a
q

Pea son equa ion (2.6) hen
!
0
sa ises
he equa ion

0
!
0
=
aq
H
!
0
(H

?
)
0
, whe e
a
=
q

. So,
!
0
= H
n
(
!
0
)
aq
(H

?
)
0

0
; a
=
q

;
o

= Log
q
(
a
)
;
whe e Log
q
deno es he p inipal loga i hm on he basis
q
,
j
q
j
<
1. In he ollowing, we will use he
no a ion 
a
=

b
aa
1
and 
a
?
=

b
a
?
a
?
1
. No ie ha , wi h his no a ion,

=
b
ax
(
x

a
1
) =
b
ax
2
+ 
ax
and

?
=
b
a
?
x
(
x

a
?
1
) =
b
a
?
x
2
+ 
a
?
x
.
(a) In his ase,
aq
(H

?
)
0

0
=
aq
2
b
a
?
a
?
1
(
a
?
1

1
q x

1)
b
aa
1
(
a

1
1
x

1)
=
aq
2

a
?
(1

a
?
1

1
q x
)

a
(1

a

1
1
x
)
:
I we ho ose now,
a
suh ha
aq
(H

?
)
0
(0) =

0
(0), i.e.,
aq
2

a
?
= 
a
, o equi alen ly,

= Log
q
(
a
) =

2 + Log
q

a

a
?
, we an apply he Lemma 2.1 o ge ,
!
0
=
!
0
(0)
(
a
?
1

1
q x
;
q
)
1
(
a

1
1
x
;
q
)
1
, whih leads, wi hou
any loss o gene ali y, o he ollowing weigh un ion (he e we supp ose ha
!
0
is on inuous and
!
0
(0)
6
= 0)
!
(
x
) =
j
x
j

(
a
?
1

1
q x
;
q
)
1
(
a

1
1
x
;
q
)
1
; 
= Log
q
(
a
) =

2 + Log
q

a

a
?
:
(2.8)
(b) In his ase,
aq
(H

?
)
0

0
=
aq
2

a
?

a
(1

a

1
1
x
)
=
)
!
0
=
!
0
(0)
1
(
a

1
1
x
;
q
)
1
; 
= Log
q
(
a
) =

2 + Log
q



a

a
?

;
so,
!
(
x
) =
j
x
j

(
a

1
1
x
;
q
)
1
; 
= Log
q
(
a
) =

2 + Log
q



a

a
?

:
Finally, in he las ase (), we ob ain
!
(
x
) =
j
x
j

(
a
?
1

1
q x
;
q
)
1
; 
= Log
q
(
a
) =

2 + Log
q



a

a
?

:
I I.
Le onside he o he ase, i.e., when

and

?
ha e a anishing e m on
x
. In his ase he e
a e wo p ossibili ies:
(i) deg

6
= deg

?
whih is di ided in wo sub ases (a)

=
b
ax
2
,

?
= 
a
?
x
, and (b)

= 
ax
,

?
=
b
a
?
x
2
, and
(ii) deg

= deg

?
, whih also is di ided in wo sub ases (a)

=
b
ax
2
,

?
=
b
a
?
x
(
x

a
?
1
),
a
?
1
6
= 0,
and (b)

=
b
ax
(
x

a
1
),

?
=
b
a
?
x
2
,
a
1
6
= 0.
In b o h ases, he me ho d used in he ase I o non- anishing o eÆien s an no b e used.
(i) In o de o sol e he p oblem o ase I I(i) we will gene alize an idea by Hake [12 ℄. Le us
dene he un ion
h
(

)
: [0
;
1
)
!
R
dened by
h
(

)
(
x
) =
p
x
log
q
x



; 
6
= 0
;
whih has he ollowing p op e y H
h

=
x

h

, o , equi alen ly,
h

(
q x
) =
x

h

(
x
), o all
x

0.
I we now dene he un ion
!
=
x

h
(1)
, hen, o he ase I I(i)a we ha e
H
!
= H
x

h
(1)
=
q

x

xh
(1)
=
q

x!
=
)
x
H
!
=
q

x
2
! ;
16
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
The Ro d igues o mula is e y use ul o nding he explii exp ession o he p olynomials
P
n
.
In a , using he o mula

?
n
(
x
) =
q

n
2

+
n
(1

q
)
n
x
n
n
X
k
=0
(

1)
k
q
k
(
k
+1)
2

nk

n
k

q
(
q
k

n
x
)
;

n
k

q
=
(
q
;
q
)
n
(
q
;
q
)
k
(
q
;
q
)
n

k
;
whe e

n
2

=
n
(
n

1)
2
, one easily ob ains
P
n
=
q

n
2

n
(1

q
)
n
x
n
n
X
k
=0
(

1)
k
q
k
(
k
+1)
2

nk

n
k

q
H
(
n
)

(
xq
k

n
)
!
(
q
n

k
x
)
!
(
x
)
;
o , equi alen ly,
P
n
=
n
(

1)
n
(1

q
)
n
x
n
n
X
k
=0
(

1)
k
q

k
2


n
k

q
H
(
n
)

(
xq

k
)
!
(
q

k
x
)
!
(
x
)
:
Now, aking in o aoun he
q

Pea son equa ion (2.6)
H
!
!
=

q
H

?
()
H

1
!
!
=
q 
?
H

1

;
we ob ain he ollowing explii exp ession o he
q

lassial p olynomials in e ms o he p olyno-
mials

and

?
:
P
n
=
n
(

1)
n
(1

q
)
n
x
n
n
X
k
=0
(

1)
k
q

k
2

+
k

n
k

q
k

1
Y
i
=0

?
(
xq

i
)
n

k

1
Y
i
=0

(
xq
i
)
:
(3.6)
This o mula is equi alen o he one ob ained in [5 , Eq. (4.14)℄, [23, Eq. (33)℄ and [2, Eq. (2.24)℄
o he
q

p olynomials in he non-uni o m la ie
x
(
s
) =

1
q
s
.
3.3.2 The hyp e geome i ep esen a ion

We s a wi h he
;
Jaobi/Jaobi amily, i.e., he ase when

=
b
a
(
x

a
1
)(
x

a
2
) and

?
=
b
a
?
(
x

a
?
1
)(
x

a
?
2
)
b
a
?
a
1
a
2
b
a
?
a
?
1
a
?
2
6
= 0. The o he ases an b e ob ained in a simila way. Then,
subs i u ing in he ab o e exp ession we nd ha he
q

lassial p olynomials b eomes
P
n
=
n
(
b
aa
1
a
2
)
n
(
x=a
1
;
q
)
n
(
x=a
2
;
q
)
n
(1

q
)
n
x
n
3
'
2
q

n
; a
?
1
x

1
; a
?
2
x

1
q
1

n
a
1
x

1
; q
1

n
a
2
x

1





q
;
b
a
?
b
a
q

n
+3
!
:
F om he las o mula i is no easy o see ha
P
n
a e p olynomials on
x
o deg ee exa ly equal
n
, hus, we will apply o he ab o e equa ion he ans o ma ions (3.2.5) and (3.2.3) gi en in [10,
page 61℄. No ie ha we an apply he ans o ma ion o mula (3.2.5) [10, page 61℄ b eause
he p olynomials

and
q 
?
ha e he same indep enden e m, and hen he ondi ion
b
aa
1
a
2
=
q
b
a
?
a
?
1
a
?
2
is ullled. So, he hyp e geome i ep esen a ion o he moni
q

lassial
;
Jaobi/Jaobi
p olynomials is
P
n
(
x
) =
a
n
2
(
a
?
1
=a
2
;
q
)
n
(
a
?
2
=a
2
;
q
)
n
(
a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
;
q
)
n
3
'
2
q

n
; a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
; x=a
2
a
?
1
=a
2
; a
?
2
=a
2





q
;
q
!
:
(3.7)
No ie ha , sine

and

?
a e in a ian wi h esp e o he hange
a
1
()
a
2
and
a
?
1
()
a
?
2
, hen
we an ob ain an equi alen hyp e geome i ep esen a ion
P
n
(
x
) =
a
n
2
(
a
?
1
=a
1
;
q
)
n
(
a
?
2
=a
1
;
q
)
n
(
a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
;
q
)
n
3
'
2
q

n
; a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
; x=a
1
a
?
1
=a
1
; a
?
2
=a
1





q
;
q
!
:
(3.8)

R.

Al a ez-Noda se and J. C. Medem
17
No ie also ha om any o he ab o e wo o mulas ollwos ha
P
n
is a p olynomial o deg ee exa ly
equal
n
. Be o e s a wi h he de ailed s udy o eah ase le us w i e ano he equi alen o m o
he
;
Jaobi/Jaobi p olynomials whih an b e ob ained applying he ans o ma ion (I I I.12) om
[10, page 241-242℄ o (3.7):
P
n
(
x
) =
q

n
2

(

a
?
2
)
n
(
a
?
1
=a
2
;
q
)
n
(
a
?
1
=a
1
;
q
)
n
(
a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
;
q
)
n
3
'
2
q

n
; a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
; a
?
1
=x
a
?
1
=a
2
; a
?
1
=a
1





q
;
q x=a
?
2
!
:
(3.9)
I we now ho ose

=
aq
(
x

1)(
bx


) and

?
=
q

2
(
x

aq
)(
x

q
), hen Theo em 2.1 and
Eq. (3.7) gi es, o he weigh un ion and he p olynomials, esp e i ely
!
(
x
) =
(
x=a; x=
;
q
)
1
(
bx=; x
;
q
)
1
; p
n
(
x
;
a; b; 
;
q
) =
(
aq
;
q
)
n
(
q
;
q
)
n
(
abq
n
+1
;
q
)
n
3
'
2
q

n
; abq
n
+1
; x
aq ; q





q
;
q
!
;
i.e., he Big
q

Jaobi p olynomials. I we now ho ose

=
q

N

1
hey b eomes he
q

Hahn p olyno-
mials
Q
n
(
x
;
a; b; N
j
q
) (usually hey a e w i en as p olynomials in
x
=
q

s
, see [13 , 18 ℄). Ob iously,
i we use ins ead o o mula (3.7) he o mulas (3.8) and (3.9) we ob ain o he ep esen a ions o
he Big
q

Jaobi p olynomials.
Fo he o he 11 ases we an do he same, subs i u e he p olynomials

and

?
in (3.6) and make
he o esp onding alula ions, bu he e we will show how, om he
q

lassial
;
Jaobi/Jaobi
p olynomials, an b e de i ed all o he ases by aking he app opia e limi s. A simila s udy ha e
b een done in [23 ℄. He e we will omple e i . We will gi e he de ails only in some sp eial diÆul "
ases o when he la i y and he au ay a e equi ed.

We on inue wi h he
q

lassial
;
Jaobi/Lague e p olynomials. To ob ain hem we ake
he limi
a
?
2
! 1
. Then,

=
b
a
(
x

a
1
)(
x

a
2
) and
q 
?
=
q
b
a
?
(
x

a
?
1
)(
x

a
?
2
) =
q
b
a
?
a
?
2
(
x

a
?
1
)(
x=a
?
2

1) =
b
aa
1
a
2
a
?
1
(
x

a
?
1
)(
x=a
?
2

1)
! 
b
aa
1
a
2
a
?
1
(
x

a
?
1
)
;
whe e he ela ion
b
aa
1
a
2
=
q
b
a
?
a
?
1
a
?
2
has b een used. In his ase and sine
lim
a
?
2
!1
(
a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
;
q
)
k
(
a
?
2
=a
2
;
q
)
k
=
q
(
n

1)
k

a
?
1
a
1

k
;
Eq. (3.7) b eomes
P
n
(
x
) =

a
1
a
2
a
?
1

n
(
a
?
1
=a
2
;
q
)
n
q

n
(
n

1)
2
'
1
q

n
; x=a
2
a
?
1
=a
2





q
;
q
n
a
?
1
=a
1
!
:
(3.10)
I we ho ose now

= (
x

1)(
x
+
b
) and

?
=
q

2

(
x

bq
), hen we ob ain he
q

Meixne
p olynomials
M
n
(
x
;
b; 
;
q
) = (


)
n
(
bq
;
q
)
n
q

n
2
2
'
1
q

n
; x
bq





q
;

q
n
+1

!
:
In his ase
!
(
x
) =
(
x=b
;
q
)
1
(

x=b;x
;
q
)
1
. Pu ing in he ab o e o mulas
b
=
q

N

1
and

=

p

1
we a i e
o he Quan um
q

K a huk p olynomials
K
q m
n
(
x
;
p; N
;
q
).

The nex amily is he
q

lassial
;
Jaobi/He mi e one. In his ase we ake he limi
a
?
1
; a
?
2
! 1
. Then,

=
b
a
(
x

a
1
)(
x

a
2
) and
q 
?
=
b
a
?
a
1
a
2
, hus (3.7) b eomes
P
n
(
x
) = (

a
2
)

n
q

n
2

2
'
0
q

n
; x=a
2
|





q
;
q
n
a
2
=a
1
!
:
(3.11)
18
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
Cho osing

= (
x

a
)(
x

1) and
q 
?
=
a
we ob ain he Al-Salam & Ca li z I I p olynomials
V
(
a
)
n
(
x
;
q
) = (

a
)
n
q


n
2

2
'
0
q

n
; x
0





q
;
q
n
a
!
;
I now

= (
x

i
)(
x
+
i
) and
q 
?
= 1, we a i e o he Dis e e
q

He mi e p olynomials I I
e
h
n
(
x
;
q
)
e
h
n
(
x
;
q
) =
i

n
2
'
0
q

n
; ix
|





q
;

q

n
!
=
x
n
2
'
1
q

n
; q

n
+1
0





q
2
;

q
2
x
2
!
;
and o he weigh un ion we ha e
!
(
x
) = (
ix;

ix
;
q
)

1
1
= (

x
2
;
q
2
)
1
=

Q
1
k
=0
(1 +
x
2
q
2
k
)


1
.

The
q

lassial
;
Lague e/Jaobi p olynomials. In his ase
a
2
! 1
. Then,

=

q
b
a
?
a
?
1
a
?
2
a

1
1
(
x

a
1
) and

?
=
b
a
?
(
x

a
?
1
)(
x

a
?
2
), hus Eq. (3.9) gi es
P
n
(
x
) = (

a
?
2
)
n
q

n
2

(
a
?
1
=a
1
;
q
)
n
2
'
1
q

n
; a
?
1
=x
a
?
1
=a
1





q
;
q x=a
?
2
!
=
a
n
1
(
a
?
1
=a
1
;
q
)
n
(
a
?
2
=a
1
;
q
)
n
3
'
2
q

n
; x=a
1
;
0
a
?
1
=a
1
; a
?
2
=a
1





q
;
q
!
:
(3.12)
The las equali y ollows om he Jakson ans o ma ion o mula (see [10, Eq. (I I I.5), page
241℄), o , di e ly, aking he limi in o mula (3.8). I we now ho ose

=

aq
(
x

1) and

?
=
q

2
(
x

aq
)(
x

q
), we ob ain he Big
q

Lague e p olynomials
p
n
(
x
;
a; 
;
q
) = (
aq
;
q
)
n
(
q
;
q
)
n
3
'
2
q

n
;
0
; x
aq ; q





q
;
q
!
= (
aq
;
q
)
n
(

q
)
n
q

n
2

2
'
1
q

n
; aq x

1
aq





q
;
x

!
:
No ie ha hey a e no hing else ha he Big
q

Jaobi when
b
= 0. He e
!
(
x
) =
(
x=a;x=
;
q
)
1
(
x
;
q
)
1
.
To his lass also b elong he
K
a
n
(
x
;
p; N
;
q
). In a hey a e Big
q

Lague e p olynomials wi h
pa ame e s
a
=
q

N

1
and

=
p
.

The
q

lassial
;
He mi e/Jaobi p olynomials. In his ase
a
1
; a
2
! 1
, hus

=
q
b
aa
?
1
a
?
2
and

?
=
b
a
?
(
x

a
?
1
)(
x

a
?
2
). Then, om Eq. (3.9) one easily nd
P
n
(
x
) =
q

n
2

(

a
?
2
)
n
2
'
1
q

n
; a
?
1
=x
0





q
;
q x=a
?
2
!
:
(3.13)
Now ho osing

=
a
and
q 
?
= (
x

1)(
x

a
), (3.13) leads o he Al-Salam & Ca li z I p olynomials
U
(
a
)
n
(
x
;
q
) = (

a
)
n
q

n
2

2
'
1
q

n
; x

1
0





q
;
x q
a
!
:
In his ase he
q

weigh un ion akes he o m
!
(
x
) = (
q x=a; q x
;
q
)
1
. I we pu
a
=

1, he he
Al-Salam & Ca li z I p olynomials b eomes he dis e e
q

He mi e p olynomials I
h
n
(
x
;
q
).
Fo he 0

amilies he si ua ion is mo e omplia e and a new pa ame e
Æ
should b e inluded.
R.

Al a ez-Noda se and J. C. Medem
19

To ob ain he 0

Bessel/Jaobi p olynomials we will ake he limi
a
1
; a
2
; a
?
2
!
0. Thus,

=
b
ax
2
and

?
=
b
a
?
(
x

a
?
1
)
x
, bu now we ha e a p oblem aking he limi in he exp ession
(
a
?
1
a
?
2
a

1
1
a

1
2
q
n

1
;
q
)
k
, so we will obliged he pa ame e s
a
1
; a
2
; a
?
2
end o ze o suh ha
a
?
2
a

1
1
a

1
2
=
q
Æ
, wi h
Æ
a xed ons an suh ha
q
Æ
=
b
a=
(
q
b
a
?
a
?
1
). Then, aking he limi in Eq. (3.7) we ob ain
P
n
(
x
) =
q

n
2

(

a
?
1
)
n
(
q
n
+
Æ

1
;
q
)
n
2
'
1
q

n
; q
n
+
Æ

1
0





q
;
q x=a
?
1
!
; q
Æ
=
b
a
q
b
a
?
a
?
1
:
(3.14)
To his lass b elongs he Al e na i e
q

Cha lie p olynomials
K
n
(
x
;
a; q
). In a , pu ing

=
ax
2
and

?
=
q

2
x
(1

x
), hus
q
Æ
=

aq
and hen
K
n
(
x
;
a
;
q
) =
(

1)
n
q

n
2

(

aq
n
;
q
)
n
2
'
1
q

n
;

aq
n
0





q
;
q x
!
:
Fo hem we ha e
!
(
x
) =
j
x
j

(
x

1
;
q
)

1
1
, whe e
q

=

a=q
.

Fo he 0

Bessel/Lague e p olynomials we ha e he limi
a
1
; a
2
; a
?
1
!
0 and
a
?
2
! 1
. Thus,

=
b
ax
2
and

?
=
b
a
?
(
x

a
?
1
)(
x

a
?
2
) =
b
a
?
a
?
2
(
x=a
?
2

1)(
x

a
?
1
) =
b
aa
1
a
2
a
?
1

1
q

1
(
x=a
?
2

1)(
x

a
?
1
).
I we now ake he limi in suh a way ha
a
?
1
a
1
a
2
=

q
Æ
we a i e o he un ion

?
=
b
aq

Æ

1
x
.
In his ase Eq. (3.9) immedia ely gi es
P
n
(
x
) =
q

n
(
n
+
Æ

1)
(

1)
n
1
'
1
q

n
0





q
;

q
n
+
Æ
x
!
; q
Æ
=

b
a

a
?
q
:
(3.15)
Now, se ing

=
x
2
and

?
=
q

2
x
, we ha e
q
Æ
=

q
and we ob ain he S iel jes-Wige p olyno-
mials
S
n
(
x
;
q
) = (

1)
n
q

n
2
1
'
1
q

n
0





q
;

xq
n
+1
!
:
He e
!
(
x
) =
p
x
log
q
x

1
.

The 0

Jaobi/Jaobi p olynomials. In his ase he limi is
a
2
; a
?
2
!
0 p o iding ha
a
?
2
=a
2
=
q
Æ
, hen

=
b
ax
(
x

a
1
),

?
=
b
a
?
x
(
x

a
?
1
) and (3.7) gi es
P
n
(
x
) =
q

n
2

(

a
?
1
)
n
(
q
Æ
;
q
)
n
(
a
?
1
=a
1
q
Æ
+
n

1
;
q
)
n
2
'
1
q

n
; a
?
1
=a
1
q
n
+
Æ

1
q
Æ





q
;
q x=a
?
1
!
; q
Æ
=
b
aa
1
q
b
a
?
a
?
1
:
(3.16)
Pu ing

=
ax
(
bq x

1) and

?
=
q

2
x
(
x

1),
q
Æ
=
aq
, hus
p
n
(
x
;
a; b
j
q
) =
(

1)
n
q

n
2

(
aq
;
q
)
n
(
abq
n
+1
;
q
)
n
2
'
1
q

n
; abq
n
+1
aq





q
;
q x
!
;
whih a e no hing else ha he Li le
q

Jaobi p olynomials. I now we ake

=
px
(1

x
),

?
=
q

2
x
(
x

q

N
) we a i e o he ollowing exp ession
K
n
(
x
;
p; N
;
q
) =
(

1)
n
q

nN
+

n
2

(

pq
N
+1
;
q
)
n
(

pq
n
;
q
)
n
2
'
1
q

n
;

pq
n

pq
N
+1





q
;
xq
N
+1
!
;
ha s ons i u es an al e na i e deni ion o he
q

K a huk p olynomials whih is equi alen o
he mo e" s anda d one jus using he ans o ma ion o mula (I I I.7) om [10, page 241℄
K
n
(
x
;
p; N
;
q
) =
(
q

N
;
q
)
n
(

pq
n
;
q
)
n
3
'
2
q

n
; x;

pq
n
q

N
;
0





q
;
q
!
:
20
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus
Finally, we ha e
!
(
x
) =
j
x
j

(
q x
;
q
)
1
(
q bx
;
q
)
1
,
q

=
a
and
!
(
x
) =
j
x
j

(
q
N
+1
x
;
q
)
1
(
x
;
q
)
1
,
q

=
pq
N
o he weigh
un ions o he Li le
q

Jaobi and
q

K a huk p olynomials, esp e i ely.

The 0

Jaobi/Lague e p olynomials. In his ase we ake he limi is
a
2
; a
?
2
!
0 and
a
?
1
! 1
in suh a way ha
a
?
2
=a
2
=

q
Æ
, so

=
b
ax
(
x

a
1
),

?
=
b
aa
1
q

Æ

1
x
= 
a
?
x
, and hen
P
n
(
x
) = (

a
1
)
n
q

n
(
n
+
Æ

1)
2
'
1
q

n
; x=a
1
0





q
;

q
n
+
Æ
!
; q
Æ
=
b
aa
1
q

a
?
:
(3.17)
Pu ing

=
ax
(
x
+ 1) and

?
=
q

2
x
, hen
q
Æ
=

aq
, and we ob ain he
q

Lague e p olynomials
L

n
(
x
;
q
)

L
n
(
x
;
a
;
q
)
L
n
(
x
;
a
;
q
) = (

1)
n
q

n
2
a

n
2
'
1
q

n
;

x
0





q
;
aq
n
+1
!
; !
(
x
) =
j
x
j

(

x
;
q
)
1
; q

=

a:
I we now ho ose

=
x
(
x

1) and

?
=
q

2
ax
, we ob ain
q
Æ
=
q =a
and hen we a i e o he
q

Cha lie p olynomials
C
n
(
x
;
a
;
q
) = (

1)
n
q

n
2
a
n
2
'
1
q

n
; x
0





q
;

q
n
+1
a
!
; !
(
x
) =
j
x
j

(
x
;
q
)
1
; q

=
a

1
:

The 0

Jaobi/Bessel p olynomials. He e we ake he limi is
a
2
; a
?
1
; a
?
2
!
0 in suh a way ha
a
?
1
a
?
2
=a
2
=
q
Æ
, so

=
b
ax
(
x

a
1
),

?
=
b
a
?
x
2
=
b
aa
1
q

Æ

1
x
2
, and hen (3.7) gi es
P
n
(
x
) =
q
n
(
n
+
Æ

1)
(
q
n
+
Æ

1
=a
1
;
q
)

1
n
2
'
0
q

n
; q
n
+
Æ

1
=a
1
|





q
;
xq
1

Æ
!
; q
Æ
=
b
aa
1
q
b
a
?
:
(3.18)
This amily do es no app ea in he
q

Askey Sheme unless hey a e no a i ial limi o a mo e
gene al
q

amily. We will ake he ollowing pa ame e iza ion

=
ax
(
x

b
) and

?
=
q

2
x
2
.
Then,
q
Æ
=
abq
and we ob ain ha his 0

Jaobi/Bessel p olynomials, deno ed by
j
n
(
x
;
a; b
)
j
n
(
x
;
a; b
) = (
ab
)
n
q
n
2
(
aq
n
;
q
)

1
n
2
'
0
q

n
; aq
n
|





q
;
x=
(
ab
)
!
; !
(
x
) =
j
x
j

(
bq =x
;
q
)
1
; q

=
a

1
q

5
:
They main da a a e shown in Table 3.3.2.

The 0

Lague e/Jaobi p olynomials. In his ase
a
2
; a
?
2
!
0,
a
1
! 1
,
q
Æ
=

a
?
2
=a
2
, hen

= 
ax
=
b
a
?
a
?
1
q
Æ
+1
x
,

?
=
b
a
?
x
(
x

a
?
1
), and
P
n
(
x
) = (

a
?
1
)
n
q

n
2

(

q
Æ
;
q
)
n
2
'
1
q

n
;
0

q
Æ





q
;
xq =a
?
1
!
; q
Æ
=

a
b
a
?
a
?
1
q
:
(3.19)
Pu ing

=

ax
and

?
=
q

2
x
(
x

1), hus
q
Æ
=

aq
, and we ob ain he Li le
q

Lague e o
Wall p olynomials
p
n
(
x
;
a
j
q
) = (

1)
n
q

n
2

(
aq
;
q
)
n
2
'
1
q

n
;
0
aq





q
;
q x
!
; !
(
x
) =
j
x
j

(
q x
;
q
)
1
; q

=

a:

Finally, he 0

Lague e/Bessel amily ollows om Eq. (3.8) aking he limi
a
1
; a
?
1
; a
?
2
!
0
and
a
2
! 1
p o iding ha
a
?
1
a
?
2
=a
1
=

q
Æ
, hus

= 
ax
=
b
a
?
q
Æ
+1
x
,

?
=
b
a
?
x
2
and
P
n
(
x
) = (

1)
n
q
n
(
n
+
Æ

1)
2
'
0
q

n
;
0
|





q
;

xq
1

Æ
!
; q
Æ
=

a
q
b
a
?
:
(3.20)
R.

Al a ez-Noda se and J. C. Medem
21
As he ase o 0

Jaobi/Bessel, his ase leads o a new amily whih is no in he
q

Askey Tableau.
In his ase we will adop he pa ame e iza ion

= 
ax
=
b
a
?
ax
,

?
=
q

2
x
2
,
q
Æ
=
aq
, hus
P
n
(
x
)

l
n
(
x
;
a
) = (

a
)
n
q
n
2
2
'
0
q

n
;
0
|





q
;

x=a
!
; !
(
x
) =
j
x
j

p
x
log
q
x

1
+1
; q

=
a=q :
Rema k 3.2
No ie ha in some examples he
q

weigh un ions looks e y die en om he
ones gi en in [13℄. Some imes he eason is he inde e mina eness o he assoia ed momen p ob-
lem (e.g. he S iel jes-Wiege polynomials o he
q

Lague e polynomials. Also, beause some imes
ins ead he
q

in eg als, dis e e sums a e used (see e.g. he example o he Li le
q

Jaobi polyno-
mials in [13℄).
Table 3.3.2: The
q

lassial p olynomials
j
n
(
x
;
a; b
) and
l
n
(
x
;
a
)
P
n
j
n
(
x
;
a; b
)
l
n
(
x
;
a
)
 ax
(
x

b
)
ax

?
q

2
x
2
q

2
x
2
abq
+(1

aq
)
x
q
(1

q
)
aq

x
(
q

1)
q
b

n

q

n
[
n
℄(
a
+
q
n
)
1

q
q

n
[
n
℄
1

q
n
q

n
2

+
n
(1

q
)
n
(
aq
n
;
q
)
n
q

n
2

+
n
(1

q
)
n
d
n
abq
n
(
1

q
n
+
aq
2
n

q
n
+1
)
(1

aq
2
n

1
)(1

aq
2
n
+1
)
aq
n

q
n
+
q
n
+1

1

g
n

a
2
b
2
q
3
n

1
(1

q
n
)
(
1

aq
n

1
)
(1

aq
2
n

1
)
2
(1

aq
2
n
)(1

aq
2
n

2
)
a
2
q
3
n

1
(
q
n

1)
a
n
a
[
n
℄ 0
b
n

ab
[
n
℄(1

aq
n
)
(
1+
aq
2
n
)
(1

aq
2
n

1
)(1

aq
2
n
+1
)
a
[
n
℄

n
a
2
b
2
q
2
n

1
[
n
℄(1

aq
n
)
(
1

aq
n

1
)
(1

aq
2
n

1
)
2
(1

aq
2
n
)(1

aq
2
n

2
)
a
2
q
2
n

1
[
n
℄
e
n
abq
n
(1

q
n
)
(
1+
aq
2
n
)
(1

aq
2
n

1
)(1

aq
1+2
n
)
aq
n
(
q
n

1)
h
n
a
3
b
2
q
4
n

2
(1

q
n
)
(
1

q
n

1
)
(1

aq
2
n

1
)
2
(1

aq
2
n
)(1

aq
2
n

2
)
0
d
0
n
abq
n
+1
(
1

q
n

q
n
+1
+
aq
2
n
+2
)
(1

aq
2
n
+1
)(1

aq
2
n
+3
)
aq
n
+1

q
n
+
q
n
+1

1

g
0
n

a
2
b
2
q
3
n
+1
(1

q
n
)
(
1

aq
n
+1
)
(1

aq
2
n
)(1

aq
2
n
+1
)
2
(1

aq
2
n
+2
)
a
2
q
3
n
+1
(
q
n

1)
No ie ha o all 0
< q <
1, he p olynomials
l
n
(
x
;
a
) ne e ons i u es a p osi i e deni e
amily sine
g
n
<
0 (see he Fa a d heo em (1.1)). The ase i he
j
n
(
x
;
a; b
) p olynomials is
mo e omplia ed. Ne e heless, ho osing
a
=
q

N
i is easy o show ha
j
n
(
x
;
a; b
) ons i u e a
ni e amily (simila o he
q

Hahn p olynomials) whih is p osi i e deni e sine
g
n
>
0 o all
n
= 0
;
1
;::: ;
[
N =
2℄. The de ailed s udy o he p osi i e deni e ases in dep endene o he o o s o

22
q

Classial polynomials and he
q

Askey and Niki o o -U a o Tableaus

and

?
will b e onside ed in a o homing pap e .
Aknowledgemen s:
This wo k has b een pa ially supp o ed by he Jun a de Andalua (FQM-
207), he Eu op ean p o je INTAS-93-219-ex and by he Spanish Di eion Gene al de Ense ~nanza
Sup e io (DGES) g an s PB-96-0120-C01-01. We hanks J. S. Dehesa and F. Ma ellan o help ull
diussions and ema ks.
Re e enes
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Complex Analysis.
MG aw-Hill, New Yo k, 1953.
[2℄ R.

Al a ez-Noda se and J. A es u, On he
q-
p olynomials in he exp onen ial la ie
x
(
s
) =

1
q
s
+

3
.
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8
(1999) (In p ess).
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
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
Al a ez-Noda se and F. Ma ellan,
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
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Al a ez-Noda se and J. C. Medem
23
[19℄ J. C. Medem and F. Ma ellan,
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
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