RECURRENCE RELATIONS FOR CONNECTION COEFFICIENTS
BETWEEN Q-ORTHOGONAL POLYNOMIALS OF DISCRETE
VARIABLES IN THE NON-UNIFORM LATTICE
x
(
s
) =
q
2
s
.
1
R.

Al a ez-No da se
2
Depa amen o de Ma ema icas. Escuela Poli ecnica Supe io .
Uni e sidad Ca los III de Mad id. Bu a que 15, 28911,Leganes,Mad id.
A. Ron eaux
3
Ma hema ical Physics, Facul es Uni e si ai es No e-Dame de la Paix,
B-5000 Namu , Belgium.
Key wo ds and ph ases: q-Hahn, q-Meixne , q-Cha lie and q-K a chuk
p olynomials, disc e e p olynomials, q-p olynomials p olynomials.
AMS (MOS, 1991) sub jec classica ion:
33D45
Abs ac
We ob ain he s uc u e ela ions o q-o hogonal p olynomials in he exp o-
nen ial la ice
q
2
s
and om ha we cons uc he ecu ence ela ion o he
connec ion co ecien s b e ween wo amilies o p olynomials b elonging o he
classical class o disc e e q-o hogonal p olynomials. An explici example is also
gi en.
1 In o duc ion.
Gi en wo amilies o Polynomials, deno ed by
P
n
(
x
) and
Q
m
(
x
), o deg ee exac ly
equal o esp ec i ely
n
and
m
, he
Connec ion P oblem
asks o compu e he so-called
Connec ion Coecien s
C
m
(
n
) dened by he ela ion:
P
n
(
x
) =
n
X
m
=0
C
m
(
n
)
Q
m
(
x
)
:
When b o h amilies a e
o hogonal
wi h esp ec o wo die en measu es he
Connec ions Co ecien s sa is y a ela i e simple ecu ence ela ion, bu mixing in
he (
m; n
) able h ee adjacen
m
and h ee adjacen
n
c ossing
a (
m; n
).
The  s su ey on his opic was gi en by Askey 20 yea s ago [1]-[2], gi ing in
some cases explici exp ession o he Co ecien s and discussing also he p osi i i y
p op e ies o hese Co ecien s.
1
Oc ob e 29, 1996
2
e-mail ena o@dulcinea .uc3m.es Fax:(+431) 6249430
3
e-mail And e.Ron [email p o ec ed] e Fax(+3281) 724707
1
I was no iced only ecen ly ha an addi ional assump ion on he o hogonal-
i y measu e gi es o
C
m
(
n
) a ecu ence only in
m; n
b eing xed. This O hogo-
nali y class is called
semi-classical
and is e y la ge [11], [7] . The classical (con-
inuous) amily:
Jacobi, Bessel, Lague e, He mi e
(see o ins ance [12]) and he
classical(disc e e) amily:
Hahn , K a chuk, Meixne , Cha lie
(see o ins ance [13])
a e o cou se included in he semi-classical class. When he o hogonali y measu e
is dened by a weigh

(
x
), he semi-classical class co e s all weigh s solu ion o a
linea  s o de die en ial (o die ence) equa ion wi h p olynomial co ecien s.
The key p op e y inside he semi-classical class, in o de o ob ain a one index
(m) ecu ence ela ion o
C
m
(
n
), comes om he exis ence o a so called S uc u e
Rela ion, linking linea ly he de i a i e (o die ence) o
P
n
(
x
) imes a p olynomial,
o a xed combina ion o
P
k
(
x
).
An algo i hm has b een gi en ecen ly building o b o h disc e e and con inuous
classical amilies (see [3], [15] and [16] ) he explici ecu ences o
C
m
(
n
), sol ing in
many cases heses ecu ences wi h he help o
Ma hema ica
[20].
Lo oking o he si ua ion o which a s uc u e ela ion is known explici ly, we eal-
ize ha , om he da a o O hogonal Polynomial on he exp onen ial la ice
x
(
s
) =
q
2
s
(a small subse o he q-wo ld). He e we need o p oin ou ha exi s wo die en
p oin o iew in he s udy o he q-p olynomials. The  s one, in he amewo k
o he q-basic hyp e geome ic se ies [6], [8], [9] and he second, in he amewo k o
he heo y o die ence equa ions de elop ed by Niki o o e al. [12], [13], [14]. In
his wo k we will use he second one b ecause i gi es us he p ossibili y o p o ide an
uni o m ea men o se e al classes o o hogonal p olynomials and, p obably, i is
he b es way o nd u he applica ions.
This pap e shows how o apply he echnique o a pa icula (simple) case: he
exp onen ial la ice, building  s he co esp onding S uc u e Rela ions.
2 S uc u e ela ions o q-o hogonal p olynomials on
he exp onen ial la ice
x
(
s
) =
q
2
s
.
Le us o s a wi h he s udy o some gene al p op e ies o o hogonal p olynomials
o a disc e e a iable in non-uni o m la ices. Le b e
~

(
x
(
s
))
4
4
x
(
s

1
2
)
5
Y
(
s
)
5
x
(
s
)
+
~

(
x
(
s
))
2

4
Y
(
s
)
4
x
(
s
)
+
5
Y
(
s
)
5
x
(
s
)

+
Y
(
s
) = 0
;
5
(
s
) =
(
s
)

(
s

1)
;
4
(
s
) =
(
s
+ 1)

(
s
)
;
(1)
he
second o de die ence equa ion o hype geome ic ype
o some la ice unc ion
x
(
s
), whe e
5
(
s
) =
(
s
)

(
s

1) and
4
(
s
) =
(
s
+ 1)

(
s
) deno e he
backwa d and o wa d ni e die ence quo ien s, esp ec i ely. He e ~

(
x
) and ~

(
x
)
a e p olynomials in
x
(
s
) o deg ee a mos 2 and 1, esp ec i ely, and

is a cons an .
2
The p e ious equa ion (1) can b e ob ained om he classical
hype geome ic equa ion
~

(
x
)
y
00
(
x
) + ~

(
x
)
y
0
(
x
) +
y
(
s
) = 0
;
ia he disc e iza ion o he  s and second de i a i es
y
0
and
y
00
in an ap opia e
la ice [12], [13]. I is b e e o ew i e (1) in he equi alen o m (see [13] and [14])

(
s
)
4
4
x
(
s

1
2
)
5
Y
(
s
)
5
x
(
s
)
+

(
s
)
4
Y
(
s
)
4
x
(
s
)
+
Y
(
s
) = 0
;

(
s
) = ~

(
x
(
s
))

1
2
~

(
x
(
s
))
4
x
(
s

1
2
)
; 
(
s
) = ~

(
x
(
s
))
:
(2)
The
q-o hogonal polynomials
P
n
(
x
(
s
))
q

P
n
(
s
)
q
on he exp onen ial la ice
x
(
s
) =
q
2
s
a e, o gi en unc ions

(
s
) and

(
s
), he p olynomial (in p owe s o
x
(
s
) =
q
2
s
)
solu ion o he second o de die ence equa ion (2).
The
k-o de die ence de i a i e
o he p olynomials
P
n
(
x
(
s
))
q
, dened by
k n
(
s
) =
4
4
x
k

1
(
s
)
4
4
x
k

2
(
s
)
:::
4
4
x
(
s
)
[
P
n
(
x
(
s
))
q
]
 4
(
k
)
[
P
n
(
x
(
s
))
q
]
;
and
x
m
(
s
) =
x
(
s
+
m
2
)
;
also sa is y he die ence equa ion o hyp e geome ic yp e o he o m

(
s
)
4
4
x
k
(
s

1
2
)

5
k n
(
s
)
5
x
k
(
s
)

+

k
(
s
)
4
k n
(
s
)
4
x
k
(
s
)
+

k
k n
(
s
) = 0
;
(3)
whe e (see [13], page 62, Equa ion (3.1.29))

k
(
s
) =

(
s
+
k
)


(
s
) +

(
s
+
k
)
4
x
(
s
+
k

1
2
)
4
x
k

1
(
s
)
;
and

k
=

n
+
k

1
X
m
=0
4

m
(
s
)
4
x
m
(
s
)
:
These p olynomial solu ions deno ed by
P
n
(
x
(
s
))
q

P
n
(
s
)
q
sa is y he o hogonali y
p op e y
b

1
X
s
i
=
a
P
n
(
x
(
s
i
))
q
P
m
(
x
(
s
i
))
q

(
s
i
)
4
x
(
s
i

1
2
) =

nm
d
2
n
;
(4)
whe e

(
x
) is some non-nega i e unc ion (weigh - unc ion), i.e.,

(
s
i
)
4
x
(
s
i

1
2
)
>
0 (
a

s
i

b

1)
;
supp o ed in a coun able subse o he eal line [
a; b
] (
a; b
can b e
1
). The unc-
ions

(
s
) and

k
(
s
) a e he solu ions o he Pea son- yp e die ence equa ions ([13],
Eq.(3.2.9) and (3.2.10) page 64)
4
4
x
(
s

1
2
)
[

(
s
)

(
s
)] =

(
s
)

(
s
)
;
(5)
3
and
4
4
x
k
(
s

1
2
)
[

(
s
)

k
(
s
)] =

k
(
s
)

k
(
s
) (6)
and

(
s
) sa is y he condi ion [14]:

(
s
)

(
s
)
x
k
(
s

1
2
)
j
s
=
a;b
= 0
;
8
k ; l
2
IN
(
IN
=
0
;
1
;
2
; :::
g
)
:
In (4)
d
2
n
deno es he squa e o he no m o he co esp onding o hogonal p olynomials.
The q-o hogonal p olynomials sa is y a h ee e m ecu ence ela ions (TTRR) o
he o m
x
(
s
)
P
n
(
s
)
q
=

n
P
n
+1
(
s
)
q
+

n
P
n
(
s
)
q
+

n
P
n

1
(
s
)
q
;
(7)
wi h he ini ial condi ions
P

1
(
s
)
q
= 0
; P
0
(
s
)
q
= 1
:
I is well known [13]-[14], ha he p olynomial solu ions o equa ion (2), deno ed by
P
n
(
x
(
s
))
q
, a e uniquely de e mined, up o a no malizing ac o
B
n
, by he die ence
analog o he Ro d igues o mula (see [13] page 66 Eq. (3.2.19) ):
P
n
(
s
)
q
=
B
n

(
s
)
5
(
n
)
n
[

n
(
s
)]
5
(
n
)
n
=
5
5
x
1
(
s
)
5
5
x
2
(
s
)
:::
5
5
x
n
(
s
)
[

n
(
s
)]
;
(8)
whe e

n
(
s
) =

(
n
+
s
)
Q
n
k
=1

(
s
+
k
)
:
These solu ions co esp ond o some alues o

n
- he eigen alues o equa ion (2), which is compu ed om ( see [13], page 104 and
[14] )

n
=

1
2
[
n
]
q
(
q
n

1
+
q

n
+1
) ~

0
+ [
n

1]
q
~

00
g
;
(9)
whe e ~

(
s
) =

(
s
) +
1
2
~

(
s
)
4
x
(
s

1
2
) and ~

(
s
) =

(
s
) (see Eq. (2)).
He e [
n
]
q
deno es he so called
q-numbe s
[
n
]
q
=
q
n

q

n
q

q

1
=
sinh (
hn
)
sinh (
h
)
; q
=
e
h
:
2.1 The  s s uc u e ela ion o he q-p olynomials in he la ice
x
(
s
) =
q
2
s
.
Le us now y o ob ain a s uc u e ela ion o he q-p olynomials in he exp o-
nen ial la ice
x
(
s
) =
q
2
s
. (Fo he linea la ice see [13] Eq.(2.2.10) page 24.)
Fi s o all, we ew i e he Ro d igues equa ion (8) in ano he o m. We will use
he linea i y o he op e a o
5
(
n
)
n
, as well as he iden i y
5
x
k
(
s
) =
q
k
5
x
(
s
)
:
Then, a s aigh o wa d calcula ion gi es us
4
P
n
(
s
)
q
=
q

n
(
n
+1)
2
B
n

(
s
)

5
5
x
(
s
)

n
[

n
(
s
)]
;

5
5
x
(
s
)

n
=
n- imes
z}| {
5
5
x
(
s
)
:::
5
5
x
(
s
)
:
(10)
Now, om o mulas (5) and (10) we nd
5

n
+1
(
s
)
5
x
n
+1
(
s
)
=
5
[

n
(
s
+ 1)

(
s
+ 1)]
5
x
n
(
s
+
1
2
)
=
4
[

(
s
)

n
(
s
)]
4
x
n
(
s

1
2
)
=

n
(
s
)

n
(
s
)
:
Then by using he Ro d igues o mula (8) we ob ain
P
n
+1
(
s
)
q
=
B
n
+1

(
s
)
5
(
n
+1)
n
+1
[

n
(
s
)] =
B
n
+1

(
s
)
5
(
n
)
n
5

n
+1
(
s
)
5
x
n
+1
(
s
)
=
=
B
n
+1

(
s
)
5
(
n
)
n
[

n
(
s
)

n
(
s
)] =
q

n
(
n
+1)
2
B
n
+1

(
s
)

5
5
x
(
s
)

n
[

n
(
s
)

n
(
s
)]
:
(11)
In o de o ob ain an exp ession o
h
5
5
x
(
s
)
i
n
[

n
(
s
)

n
(
s
)] we successi ely apply he
o mula
5
(
s
)
g
(
s
) =
(
s
)
5
g
(
s
) +
g
(
s

1)
5
(
s
), as well as o mulas
4

n
(
s
)
4
x
(
s
)
=
q
n

0
n
;

5
5
x
(
s

1)

n
=
q
2
n

5
5
x
(
s
)

n
:
Then, Eq. (11) gi es us he ollowing
P
n
+1
(
s
)
q
=
q

n
(
n
+1)
2
B
n
+1

(
s
)



n
(
s
)

5
5
x
(
s
)

n
[

n
(
s
)] +
q
2
n

1
[
n
]
q

0
n

5
5
x
(
s
)

n

1
[

n
(
s

1)]
!
:
(12)
Using he Ro d igues o mula o he die ence de i a i e o he p olynomial ([13],
Eq. (3.2.18) page 66) we nd (no ice ha
4
x
(
s

1) =
q

2
4
x
(
s
)):
5
P
n
(
s
)
q
5
x
(
s
)
=
4
P
n
(
s

1)
q
4
x
(
s

1)
=

q

(
n

1)(
n
+2)
2

n
B
n

(
s
)

(
s
)

5
5
x
(
s

1)

n

1
[

n
(
s

1)] =
=

q

(
n

1)(
n

2)
2

n
B
n

(
s
)

(
s
)

5
5
x
(
s
)

n

1
[

n
(
s

1)]
:
The e o e, equa ion (12) can b e ew i en in he o m
P
n
+1
(
s
)
q
=
B
n
+1

n
(
s
)
B
n
P
n
(
s
)
q

[
n
]
q
B
n
+1

0
n

(
s
)

n
B
n
5
P
n
(
s
)
q
5
x
(
s
)
and hen, he ollowing
die en ia ion o mula
holds

(
s
)
5
P
n
(
s
)
q
5
x
(
s
)
=

n
[
n
]
q

0
n


n
(
s
)
P
n
(
s
)
q

B
n
B
n
+1
P
n
+1
(
s
)
q

:
(13)
5

I we now use he p owe expansion o

n
(
s
), i.e.,

n
(
s
) =

0
n
x
n
(
s
) +

n
(0) =

0
n
q
n
x
(
s
) +

n
(0) and he TTRR (7) we ob ain he
 s s uc u e ela ion

(
s
)
5
P
n
(
s
)
q
5
x
(
s
)
=
~
S
n
P
n
+1
(
s
)
q
+
~
T
n
P
n
(
s
)
q
+
~
R
n
P
n

1
(
s
)
q
;
(14)
whe e
~
S
n
=

n
[
n
]
q

q
n

n

B
n

0
n
B
n
+1

;
~
T
n
=

n
[
n
]
q

q
n

n


n
(0)

0
n

;
~
R
n
=

n
q
n

n
[
n
]
q
:
(15)
2.2 The second s uc u e ela ion o he q-p olynomials in he la -
ice
x
(
s
) =
q
2
s
.
Le us y o ob ain now he second s uc u e ela ion. Fi s ly, we no ice ha
4
5
P
n
(
s
)
q
5
x
(
s
)
=
4
P
n
(
s
)
q
4
x
(
s
)

5
P
n
(
s
)
q
5
x
(
s
)
:
Then, by using he die ence equa ion (2)

(
s
)
5
P
n
(
s
)
q
5
x
(
s
)
=

(
s
)
4
P
n
(
s
)
q
4
x
(
s
)


(
s
)
4
5
P
n
(
s
)
q
5
x
(
s
)
=
= [

(
s
) +

(
s
)
4
x
(
s

1
2
)]
4
P
n
(
s
)
q
4
x
(
s
)
+

n
4
x
(
s

1
2
)
P
n
(
s
)
q
:
and (14) we nd
[

(
s
) +

(
s
)
4
x
(
s

1
2
)]
4
P
n
(
s
)
q
4
x
(
s
)
=
~
S
n
P
n
+1
(
s
)
q
+
+(
~
T
n


n
4
x
(
s

1
2
))
P
n
(
s
)
q
+
~
R
n
P
n

1
(
s
)
q
;
(16)
Now, aking in o accoun ha
4
x
(
s

1
2
) = (
q

q

1
)
x
(
s
), and using he TTRR (7)
we nally ob ain he
second s uc u e ela ion
[

(
s
) +

(
s
)
4
x
(
s

1
2
)]
4
P
n
(
s
)
q
4
x
(
s
)
=
S
n
P
n
+1
(
s
)
q
+
T
n
P
n
(
s
)
q
+
R
n
P
n

1
(
s
)
q
;
(17)
whe e
S
n
=
~
S
n

(
q

q

1
)

n

n
;
T
n
=
~
T
n

(
q

q

1
)

n

n
;
R
n
=
~
R
n

(
q

q

1
)

n

n
:
(18)
6
3 Recu ence ela ions o connec ion co ecien s.
Le us conside wo amilies o q-p olynomials
P
n
(
x
) and
Q
n
(
x
) b elonging o he
class o disc e e o hogonal p olynomials in he exp onen ial la ice
x
(
s
) =
q
2
s
. Each
p olynomial
P
n
(
x
) can b e ep esen ed as a linea combina ion o he p olynomials
Q
n
(
x
). In pa icula
P
n
(
x
) =
n
X
m
=0
C
m
(
n
)
Q
m
(
x
)
:
(19)
Fo he amily
P
n
(
x
) we will use he no a ion
1.

(
s
),

(
s
) and

n
o he die ence equa ion (2)
2.

n
,

n
and

n
o he TTRR (7) co ecien s
3.
S
n
,
R
n
and
T
n
o he second s uc u e ela ion (17)
and o he
Q
n
(
x
)
1. 

(
s
), 

(
s
) and


n
o he die ence equa ion (2)
2. 

n
,


n
and 

n
o he TTRR (7) co ecien s
3.

S
n
,

R
n
and

T
n
o he second s uc u e ela ion (17)
Since he p olynomials o he amily
P
n
(
x
) a e solu ions o he second o de die ence
equa ion (2) he ac ion o he die ence op e a o o second o de
^
L
, dened by
^
L
=

(
s
)
4
4
x
(
s

1
2
)

5
5
x
(
s
)

+

(
s
)
4
4
x
(
s
)
+

n
;
on Eq. (19) gi es us
n
X
m
=0
C
m
(
n
)
"

(
s
)
4
4
x
(
s

1
2
)

5
Q
m
(
x
)
5
x
(
s
)

+

(
s
)
4
Q
m
(
x
)
4
x
(
s
)
+

n
Q
m
(
x
)
#
= 0
:
(20)
Mul iplying by 

(
s
) and using


(
s
)
4
4
x
(
s

1
2
)

5
Q
m
(
x
)
5
x
(
s
)

=



(
s
)
4
Q
m
(
x
)
4
x
(
s
)



n
Q
m
(
x
)
;
we ob ain he ela ion
n
X
m
=0
C
m
(
n
)

(

(
s
) 

(
s
)



(
s
)

(
s
))
4
Q
m
(
x
)
4
x
(
s
)
+ (

n


(
s
)


(
s
)


m
)
Q
m
(
x
)

= 0
:
(21)
In o de o elimina e
4
Q
m
(
x
)
4
x
(
s
)
, we mul iply (21) by 

(
s
) + 

(
s
)
4
x
(
s

1
2
) and use he
second s uc u e ela ion (17) o he
Q
m
(
x
) amily, ob aining
n
X
m
=0
C
m
(
n
)

(

(
s
) 

(
s
)



(
s
)

(
s
))(

S
m
Q
m
+1
(
x
) +

R
m
Q
m

1
(
x
) +

T
m
Q
m
(
x
))+
+ ( 

(
s
) + 

(
s
)
4
x
(
s

1
2
))(

n


(
s
)


(
s
)


m
)
Q
m
(
x
)

= 0
:
(22)
7
The las s ep consis s o expand he emaining e ms o yp e 

2
(
s
)
Q
m
(
x
), 

(
s
)

(
s
)
Q
m
(
x
),

(
s
) 

(
s
)
Q
m
(
x
) and 

(
s
)

(
s
)
Q
m
(
x
) in linea combina ion o
Q
m
(
x
) by using he
TTRR (7) ep ea edly o he
Q
m
(
x
) amily.
A e his p o cess, (22) educes o
N
X
m
=0
M
m
[
C
0
(
n
)
; C
1
(
n
)
; :::; C
n
(
n
)]
Q
m
(
x
) (23)
whe e
N
=
max
n
+
deg 
+
deg
( 

)
; n
+ 2
deg
( 

)
; n
+ 1 +
deg
( 

) +
deg
(

)
; n
+ 1 +
deg
( 

) +
deg
(

)
;
1 +
deg
( 

) +
deg
( 

)
g
:
Taking in o accoun he linea indep endence o he amily
Q
m
(
x
) we ob ain he
linea sys em
M
m
[
C
0
(
n
)
; C
1
(
n
)
; :::; C
n
(
n
)] = 0
:
(24)
These ela ions con ain (linea ly) se e al connec ion co ecien s
C
i
(
n
) dep ending es-
sen ially on he deg ees o

(
s
) and 

(
s
). In he mos gene al si ua ion hey a e
p olynomials o second deg ee in
x
(
s
) =
q
2
s
. In his case we ob ain a ela ion o he
ollowing yp e he linea sys em we a e lo oking o
M
m
[
C
m
+4
(
n
)
; :::; C
m

4
(
n
)] = 0
;
(25)
which is alid o
n
g ea e o equal han he numb e o ini ial condi ions needed
o s a he ecu sion (
n

8). No ice ha o (
n <
8) he sys em also gi es he
solu ion, bu no in a ecu en way.
No ice ha o he q-Hahn, q-Meixne , q-Cha lie and q-K a chuk p olynomials,
as i is show in [13], able
3.3
, page 95, he

(
s
) is a p olynomial o second deg ee in
x
(
s
) =
q
2
s
. This implies ha o such p olynomials he ecu ence ela ions o he
connec ion co ecien all a e o he o m (25). Again we wan o ema k ha we a e
ollow he no a ion in o duced by Niki o o e al. [13].
4 Recu ence ela ions o connec ion co ecien s: A
simple example.
As we ha e no iced in he p e ious sec ion he ecu ence ela ion o connec ion
co ecien s o die en classes o q-p olynomials a e o o la ge (8- e ms). He e we will
analyze a mo e simple case. Fi s ly, no ice ha in he p e ious algo i hm we ha e
no used he o hogonali y p op e y o he p olynomials
P
n
, and only ha hey sa is y
a die ence equa ion. On he o he hand, o he p olynomials
Q
m
we need o ha e
s uc u e ela ions as well as h ee e m ecu ence ela ions. Le us o show and
example in which we will decomp ose a se o p olynomials
P
n
(
s
), sa is ying a ce ain
die ence equa ion o  s o de in he la ice
x
(
s
) =
q
2
s
, as a linea combina ion o
he o hogonal q-p olynomials dened in he same la ice, i.e., he q-Hahn, q-Meixne ,
q-K a chuk and q-Cha lie o hogonal p olynomials (see [13], [4] and [17])
8
Le us dene he quan i ies (
s
)
q
and (
s
n
)
q
, dened by
(
s
)
q
=
q
2
s

1
q
2

1
=
q
s

1
[
s
]
q
(26)
and
(
s
n
)
q
= (
s
)
q
(
s

1)
q

(
s

n
+ 1)
q
=
n

1
Y
k
=0
q
2
s
+2
k

1
q
2

1
(27)
The quan i ies (
s
n
)
q
a e closely ela ed o he
q-S i ling numbe s
~
S
q
2
(
n; k
)
; s

q
2
(
n; k
)
[21] by o mulas
(
s
)
n
q
=
n
X
k
=0
~
S
q
2
(
n; k
)(
s
k
)
q
;
(
s
n
)
q
=
n
X
k
=0
s

q
2
(
n; k
)(
s
)
k
q
(28)
and sa is y he ollowing wo die ence equa ions (he e, as b e o e,
x
(
s
) =
q
2
s
)
(
q
2
s

1)
5
(
s
n
)
q
5
x
(
s
)

q

n
+1
[
n
]
q
(
s
n
)
q
= 0 (29)
and
(
q
2
s

2
n
+2

1)
4
(
s
n
)
q
4
x
(
s
)

q

n
+1
[
n
]
q
(
s
n
)
q
= 0
:
(30)
Since (
s
n
)
q
is a p olynomial in
x
(
s
) =
q
2
s
, i can b e ep esen ed as a linea combina ion
o he p olynomials
Q
m
(
x
), he q-p olynomials in he exp onen ial la ice. In pa icula
(
s
n
)
q
=
n
X
m
=0
C
m
(
n
)
Q
m
(
x
)
:
(31)
Le us o ob ain he ecu ence ela ion o he connec ion co ecien s
C
m
(
n
) b e ween
he (
s
n
)
q
and he q-Cha lie , q-Meixne o q-K a chuk. (Fo q-Hahn p olynomials we
will conside i sepa a ely). In o de o do ha we apply he op e a o
~
L
= (
q
2
s

1)
5
5
x
(
s
)

q

n
+1
[
n
]
q
(32)
o b o h sides o (31). Using o mula (29) (
~
L
(
s
n
)
q
= 0) and mul iplying by
q
2
s
we
ob ain he ollowing exp ession
0 =
n
X
m
=0
C
m
(
n
)

q
2
s
(
q
2
s

1)
5
Q
m
(
x
)
5
x
(
s
)

q

m
+1
[
m
]
q
q
2
s
Q
m
(
x
)

:
(33)
Taking in o accoun ha o q-Cha lie , q-Meixne and q-K a chuk he

(
s
) unc ion
in (2) coincide wi h
q
2
s
(
q
2
s

1) and applying he s uc u e ela ion (14) and he
TTRR (7) o he p e ious exp ession we nd
0 =
n
X
m
=0
C
m
(
n
)
A
m
Q
m
+1
(
x
) +
B
m
Q
m
(
x
) + 
m
Q
m

1
(
x
)
g
;
om whe e we ob ain he ollowing TTRR o he connec ion co ecien s
C
m
(
n
)
A
m

1
C
m

1
(
n
) +
B
m
C
m
(
n
) + 
m
+1
C
m
+1
(
n
) = 0
;
(34)
9