hypergeometri type on nonuniform latties
R.
Alvarez-Nodarse yz
and R.S. Costas-Santos y
y
Departamento de AnalisisMatematio.
Universidad de Sevilla. Apdo. 1160, E-41080Sevilla
z
Instituto CarlosI de Fsia Teoria y Computaional,
Universidad de Granada, E-18071Granada, Spain
24th May 2001
Abstrat
We study the fatorization of the hypergeometri-type dierene equation of
NikiforovandUvarovonnonuniform latties. An expliitformoftheraisingand
loweringoperatorsisderivedandsomerelevantexamplesaregiven.
1 Introdution
Inthispaperwewilldealwiththeso-alledfatorizationmethod(FM)ofthe
hypergeo-metri-type diereneequations onnonuniform latties. The FMwasalready usedby
Darboux[14 ℄ and Shrodinger[25 , 26 ℄to obtainthesolutionsof dierentialequations,
and also by Infeld and Hull [16℄ for nding analytial solutions of ertain lasses of
seondorder dierentialequations. Later on,Millerextendedittodiereneequations
[20 ℄ and q-dierenes{in theHahn sense{[21 ℄. Formore reent workssee e.g. [4 , 10 ,
11 , 29,30 ℄and referenestherein.
ThelassialFMwasbased ontheexisteneofa so-alledraisingand lowering
op-eratorsfortheorrespondingequationthatallowstondtheexpliitsolutionsinavery
easyway. Goingfurther,Atakishiyevandoauthors[4 ,6,10 ℄havefoundthedynamial
symmetry algebrarelatedwith theFMand thedierentialordierene equations. Of
speialinterestwasthepaperbySmirnov[27 ℄ inwhihtheequivaleneof theFMand
the Nikiforov et all theory [23℄ was shown, furthermore this paper pointed out that
theaforementionedequivaleneremainsvalidalsoforthenonuniformlattiesthatwas
shown later on in [28, 29 ℄. In partiular, in [29 ℄ a detailed study of the FM and its
equivalenewiththeNikiforovetal. approahtodiereneequations[23℄havebeen
es-tablished. Also,in[12 ℄,aspeialnonuniformlattiewasonsidered. Infat,in[12 ℄the
author onstruted the FMforthe Askey{Wilsonpolynomials usingbasiallythe
dif-fereneequationforthepolynomials. Inthepresentpaperwewillontinuetheresearh
of the nonuniformlattie ase. In fat, followingthe idea by Bangerezako [12℄ forthe
Askey{Wilson polynomials and Lorente [18 ℄ for the lassial ontinuous and disrete
ases,wewillobtaintheFMforthegeneralpolynomialsolutionsofthehypergeometri
diereneequationonthegeneralquadratinonuniformlattiex(s)=
1 q
s
+
2 q
s
+
3 .
orresponding normalized funtions whih is more natural and useful. In suh a way,
the method proposed here is thegeneralization of [12 ℄ and [18℄ to the aforementioned
nonuniform lattie.
The strutureof the paperis asfollows. In Setion2 we present some well-known
results on orthogonal polynomials on nonuniform latties [7 , 23 , 24 ℄, in setion 3 we
introdue the normalized funtions and obtain some of their properties suh as the
lowering and raising operator that allow us, in Setion 4, to obtain the fatorization
fortheseond order diereneequationsatisedbysuhfuntions. Finally,inSetion
5,some relevant examples areworked out.
2 Some basi properties of the q-polynomials
Here,we willsummarizesomeofthepropertiesoftheq-polynomialsusefulfortherest
of thework. Forfurtherinformationseee.g. [23 ℄.
Wewilldeal herewith theseond order dierene equation of the hypergeometri type
(s)
x(s 1
2 )
ry(s)
rx(s)
+(s) y(s)
x(s)
+y(s)=0;
(s)=~(x(s)) 1
2 ~
(x(s))x(s 1
2
); (s)=~(x(s));
(1)
whererf(s)=f(s) f(s 1) and f(s)=f(s+1) f(s) denotethebakwardand
forward nitedierenederivatives, respetively, (x(s))~ and ~(x(s)) arepolynomials
in x(s) of degree at most2 and 1, respetively, and is a onstant. In the following,
we willuse the following notation forthe oeÆients inthe powerexpansions in x(s)
of (s)~ and~(s)
~
(s)[x(s)℄~ = ~ 00
2 x
2
(s)+~ 0
(0)x(s)+(0);~ ~(s)~[x(s)℄=~ 0
x(s)+~(0): (2)
An important property of the above equation is that thek-order dierene derivative
of asolutiony(s) of(1), dened by
y
k (s)
q =
x
k 1 (s)
x
k 2 (s)
:::
x(s)
y(s) (k)
y(s);
also satisesa diereneequation of thehypergeometritype
(s)
x
k (s
1
2 )
ry
k (s)
q
rx
k (s)
+
k (s)
y
k (s)
q
x
k (s)
+
k y
k (s)
q
=0; (3)
where x
k
(s)=x(s+ k
2
) and[23,page 62,Eq. (3.1.29)℄
k (s)=
(s+k) (s)+(s+k)x(s+k 1
2 )
x
k 1 (s)
;
k =+
k 1
X
m=0
m (s)
x
m (s)
: (4)
It isimportant tonotiethattheabove diereneequationshave polynomialsolutions
of thehypergeometritype i x(s)isa funtionof theform[7, 24 ℄
x(s)=
1 (q)q
s
+
2 (q)q
s
+
3
(q)=
1 (q)[q
s
+q s
℄+
3
where
1 ,
2 ,
3
and q =
2
areonstants whih,ingeneral, dependon q [23,24 ℄. For
theabove lattie,astraightforwardalulationshowsthat
k
(s)isapolynomialofrst
degree inx
k
(s) of theform (see e.g. [7℄)
k
(s)=~ 0
k x
k (s)+~
k
(0); ~ 0
k =[2k℄
q ~ 00
2 +
q (2k)~
0
;
~
k (0)=
3 e 00
2 (2[k℄
q [2k℄
q )+e
0
(0)[k℄
q +
3
0
(
q
(k)
q
(2k))+~(0)
q (k);
(6)
where theq-numbers [k℄
q and
q
(k) aredenedby
[k℄
q =
q k
2
q k
2
q 1
2
q 1
2
;
q (k)=
q k
2
+q k
2
2
; (7)
and [n℄
q
!aretheq-fatorials[n℄
q !=[1℄
q [2℄
q [n℄
q .
Bothdierene equations(1)and (3)an be rewritteninthesymmetri form
x(s 1
2 )
(s)(s) ry(s)
rx(s)
+
n
(s)y(s)=0;
and
x
k (s
1
2 )
(s)
k (s)
ry
k (s)
rx
k (s)
+
k
k (s)y
k
(s)=0;
where (s) and
k
(s) are the weight funtions satisfying the Pearson-type dierene
equations
4
x(s 1
2 )
[(s)(s)℄=(s)(s);
4
x
k (s
1
2 )
[(s)
k
(s)℄=
k (s)
k (s);
(8)
respetively. In [23 ℄ it isshownthat thepolynomialsolutions of (3)(and sothe
poly-nomial solutionsof(1))are determinedbytheq-analogue oftheRodriguesformulaon
thenonuniform latties
x
k 1 (s)
x(s) P
n (x(s))
q
(k)
P
n (x(s))
q =
A
n;k B
n
k (s)
r (n)
k
n
(s); (9)
where
r (n)
k
f(s)= r
rx
k+1 (s)
r
rx
k+2 (s)
r
rx
n (s)
f(s):
A
n;k =
[n℄
q !
[n k℄
q !
k 1
Y
m=0
q
(n+m 1)e 0
+[n+m 1℄
q e 00
2
(10)
Thus[23, page66, Eq. (3.2.19)℄
P
n (x(s))
q =
B
n
(s) r
(n)
n
(s); r (n)
r
rx
1 (s)
r
rx
2 (s)
r
rx
n (s)
; (11)
where
n
(s)=(s+n) n
Y
k=1
(s+k) and
n
= [n℄
q
q
(n 1)e 0
+[n 1℄
q e 00
2
proven [23℄, by using the dierene equation of hypergeometri-type (1), that if the
boundaryondition
(s)(s)x k
(s 1
2 )
s=a;b
=0; 8k0; (13)
holds, thenthepolynomialsP
n (s)
q
areorthogonal,i.e.,
b 1
X
s=a P
n (x(s))
q P
m (x(s))
q
(s)x(s 1
2 )=Æ
nm d
2
n
; s=a;a+1;:::;b 1;
(14)
where (s) is a solution of the Pearson-type equation (8). In the speial ase of the
linear exponential lattie x(s)=q s
the above relation an be written interms of the
Jakson q-integral(see e.g. [15 , 17 ℄) R
z
2
z
1 f(t)d
q
t, denedby
Z
z2
z
1 f(t)d
q t=
Z
z2
0
f(t)d
q t
Z
z1
0
f(t)d
q t;
where
Z
z
0 f(t)d
q
t=z(1 q) 1
X
k=0 f(zq
k
)q k
; 0<q <1;
asfollows:
Z
q b
q a
P
n (t)
q P
m (t)
q !(t)d
q t=Æ
nm q
1=2
d 2
n
; t=q s
; !(t)!(q t
)=(t):
(15)
Notiethat the above boundary ondition(13) is validfork =0. Moreover, ifwe
assume thataisnite, then(13)is fullledat s=aprovidingthat (a)=0[23 , x3.3,
page70℄. Inthefollowingwe willassumethat thisonditionholds. Thesquarednorm
in(14) isgiven by[23 , Chapter3, Setion3.7.2,pag. 104℄
d 2
n
=( 1) n
A
n;n B
2
n b n 1
X
s=a
n (s)x
n (s
1
2 ):
There is also a so-alled ontinuous orthogonality. In fat, if there exist a ontour
suhthat
Z
[(z)(z)x k
(z 1
2
)℄dz=0; 8k0; (16)
then [23 ℄
Z
P
n (x(z))
q P
m (x(z))
q
(z)x(z 1
2
)dz=0; n6=m:
A simpleonsequene of theorthogonalityis thefollowingthree termreurrene
rela-tion:
x(s)P
n (x(s))
q =
n P
n+1 (x(s))
q +
n P
n (x(s))
q +
n P
n 1 (x(s))
q
; (17)
where
n ,
n and
n
areonstants. IfP
n (s)
q =a
n x
n
(s)+b
n x
n 1
(s)+ ;thenusing
(17) we nd
n =
a
n
a
n+1
;
n =
b
n
a
n b
n+1
a
n+1
;
n =
a
n 1
a
n d
2
n
d 2
n 1
: (18)
Toobtaintheexpliitvaluesof
n ,
n
wewillusethefollowinglemma,{interestingin
(k) x n (s)= [n℄ q ! [n k℄ q ! x n k k (s)+ 3 n [n 1℄ q !
[n k 1℄
q ! (n k) [n℄ q ! [n k℄ q ! x n k 1
k
(s)+:
In thease k=n 1,it beomes
(n 1)
x n
(s)=[n℄
q !x n 1 (s)+ 3 [n 1℄ q
!( n [n℄
q ):
(19)
Now, usingtheRodriguesformula(9) fork=n 1,
(n 1) P n (x(s)) q = A n;n 1 B n n 1 (s) r (n) n 1 n (s)= A n;n 1 B n n 1 (s) r rx n (s) n (s);
as well as the identities
n
(s) =
n 1
(s+1)(s+1), x
n
(s) = x
n 1 (s+
1
2
) and the
Pearsonequation (8)for
n 1
(s), we nd
(n 1) P n (x(s)) q =A n;n 1 B n n 1 (s): Thus a n = A n;n 1 B n ~ 0 n 1 [n℄ q ! =B n n 1 Y k=0 q
(n+k 1)~ 0
+[n+k 1℄
q ~ 00 2 ; and b n a n = [n℄ q ~ n 1 (0) ~ 0 n 1 + 3 ([n℄ q n): So n = B n B n+1 q (n 1)~ 0 +[n 1℄ q ~ 00 2 ( q
(2n 1 )~ 0 +[2n 1℄ q ~ 00 2 )( q (2n )~ 0 +[2n℄ q ~ 00 2 ) = B n B n+1 n [n℄ q [2n℄ q 2n
[2n+1℄
q 2n+1 and n = [n℄ q ~ n 1 (0) ~ 0 n 1
[n+1℄
q ~ n (0) ~ 0 n + 3 ([n℄ q
+1 [n+1℄
q ):
UsingtheRodrigues formula thefollowingdierene-reurrent relationfollows [1 ,23 ℄
(s) rP n (x(s)) q rx(s) = n [n℄ q 0 n n (s)P n (x(s)) q B n B n+1 P n+1 (x(s)) q ; where n
(s) is given by(6), wheretheidentity~ 0
n =
2n+1
[2n+1℄
q
hasbeenused.
Then,usingtheexpliitexpressionfortheoeÆient
n
,we nd
(s) rP n (x(s)) q rx(s) = n [n℄ q n (s) 0 n P n (x(s)) q n 2n [2n℄ q P n+1 (x(s)) q : (20)
This equation denes a raising operator in terms of the bakward dierene in the
sense that we an obtain the polynomialP
n+1
of degree n+1 from the lower degree
polynomialP
n .
From theabove equation andusingtheidentityr= r,theseond order
dier-eneequationandthethreetermsreurrenerelationwend[1 ℄lowering-typeoperator:
to thease
(s)=A 4
Y
i=1 [s s
i ℄
q =Cq
2s 4
Y
i=1 (q
s
q si
); A;C ;notvanishingonstants (22)
and hasthe form[24℄
P
n (s)
q =D
n 4
3
q n
;q
2+n 1+ P
4
i=1 si
;q s1 s
;q s1+s+
q s
1 +s
2 +
;q s
1 +s
3 +
;q s
1 +s
4 +
;q;q
; (23)
whereD
n
isanormalizingonstantandthebasihypergeometriseries
p
q
aredened
by[17℄
r
p
a
1 ;:::;a
r
b
1 ;:::;b
p ;q;z
= 1
X
k =0 (a
1 ;q)
k (a
r ;q)
k
(b
1 ;q)
k (b
p ;q)
k z
k
(q;q)
k h
( 1) k
q k
2 (k 1)
i
p r+1
;
and
(a;q)
k =
k 1
Y
m=0
(1 aq m
); (24)
is the q-analogue of the Pohhammer symbol. Instanes of suh polynomials are the
Askey{Wilson polynomials, the q-Raah polynomials and big q-Jaobi polynomials
among others[17 , 24 ℄.
3 The orthonormal funtions on nonuniform latties
In this setion we will introduea set of orthonormal funtions whih are orthogonal
withrespet to theunitweight [6 , 27 ℄
'
n (s)=
p
(s)=d 2
n P
n (x(s))
q
; (25)
e.g. forthease of disreteorthogonalitywe have
b 1
X
si=a '
n (s
i )'
m (s
i )x(s
i 1
2 )=Æ
nm :
Next,we willestablishseveral important propertiesof suh funtionswhih
gener-alize, tothenonuniformlatties, theonesgivenin[18 ℄. Inthefollowingwewillusethe
notation (s)=(s)+(s)x(s 1
2 ).
Firstof all, inserting(25) into (1), (17), (20), (21) we obtain that they satisfythe
followingdierene equation:
p
(s)(s+1) 1
x(s) '
n
(s+1)+ p
(s 1))(s) 1
rx(s) '
n (s 1)
(s)
x(s) +
(s)
rx(s)
'
n
(s)+
n x(s
1
2 )'
n
(s)=0;
(26)
thethree termreurrene relation:
n d
n+1
d
n '
n+1 (s)+
n d
n 1
d
n '
n 1
(s)+(
n
x(s))'
n
L +
(s;n)'
n
(s)=
n
2n
[2n℄
q d
n+1
d
n '
n+1
(s); (28)
and thelowering-type formula:
L (s;n)'
n
(s)=
n
2n
[2n℄
q d
n 1
d
n '
n 1
(s); (29)
where the raising-type operator L +
(s;n) and the lowering-type operator L (s;n) are
given by
L +
(s;n)
n
[n℄
q
n (s)
0
n
(s)
rx(s)
I+ p
(s 1)(s) 1
rx(s)
E ; (30)
and
L (s;n)
n
[n℄
q
n (s)
0
n +
n x(s
1
2 )+
2n
[2n℄
q
(x(s)
n )
(s)
x(s)
I
+ p
(s)(s+1) 1
x(s) E
+
;
(31)
respetively. IntheaboveformulasE f(s)=f(s 1),E +
f(s)=f(s+1)and I isthe
identityoperator.
Notie that the last two formulas have a remarkable property of giving all the
solutions'
n
(s). Infat,from(31)settingn=0andtakingintoaount that'
1 (s)
0 we an obtain'
0
(s). Then, substituting the obtained funtion in(30), we an nd
all thefuntions '
1
(s),...,'
n
(s),....
Proposition 3.1 The raising and lowering operators (30) and (31) are mutually
ad-joint.
Proof: The proof is straightforward. In fat using theboundary onditionand after
some alulationsweobtain, inthease of disreteorthogonality,theexpression
b 1
X
s
i =a
'
n+1 (s
i )
[2n℄
q
2n L
+
(s
i ;n)'
n (s
i )
x(s
i 1
2 )
= b 1
X
si=a
[2n+2℄
q
2n+2 L (s
i
;n+1)'
n+1 (s
i )
'
n (s
i )x(s
i 1
2 )=
n d
n+1
d
n :
The other ases an bedoneinan analogousway.
Proposition 3.2 The operator orresponding to the eigenvalue
n
in (26) is self
ad-joint.
Proof: Again we willprove theresult inthe ase ofdisrete orthogonality. Usingthe
orthogonalityonditions (a)(a)=(b)(b)=0(whih isa onsequene of(13)), we
an write
b 1
X
si=a '
n (s
i )
p
(s
i
1)(s
i )
1
rx(s
i )
'
l (s
i
1)x(s
i 1
2 )
= b 2
X
s 0
i =a 1
'
n (s
0
i +1)
q
(s 0
i )(s
0
i +1)
1
rx(s 0
i +1)
'
l (s
0
i )x(s
0
i +
1
= X
s
i =a
'
n (s
i +1)
p
(s
i )(s
i +1)
1
rx(s
i +1)
'
l (s
i )x(s
i +
1
2 )+
'
n (a)
p
(a 1)(a) 1
rx(a) '
l
(a 1)x(a 1
2 )
'
n (b)
p
(b 1)(b) 1
rx(b) '
l
(b 1)x(b 1
2 );
where inthelast two sumswe rst take theoperationsand r, and then substitute
theorresponding value: e.g. x(a)=x(a+1) x(a).
Now, we use the fat that '
n (s) =
p
(s)=d 2
n P
n (x(s))
q
, as well as the boundary
onditions (a)(a)=(b)(b)=0,so
p
(a 1)(a)'
n (a)'
l
(a 1)= p
(b 1)(b)'
n (b)'
l
(b 1)=0:
The other termsan betransformed ina similarway. Alltheseyield theexpression
b 1
X
s
i =a
'
l (s
i )
p
(s
i )(s
i +1)
1
x(s
i )
'
n (s
i
+1)x(s
i +
1
2 )+
p
(s
i
1)(s
i )
1
rx(s
i )
'
n (s
i
1)x(s
i 1
2 )
=
= b 1
X
s
i =a
'
n (s
i )
p
(s
i )(s
i +1)
1
x(s
i )
'
l (s
i
+1)x(s
i +
1
2 )+
p
(s
i
1)(s
i )
1
rx(s
i )
'
l (s
i
1)x(s
i 1
2 )
;
from wherethepropositioneasily follows.
4 Fatorization of dierene equation of hypergeometri
type on the nonuniform lattie
Wewilldene from(26) thefollowingoperator
H(s;n) p
(s 1)(s) 1
rx(s)
E +
p
(s)(s+1) 1
x(s) E
+
(s)
x(s) +
(s)
rx(s)
n x(s
1
2 )
I:
Clearly,theorthonormal funtions satisfy
H(s;n)'
n
(s)=0:
Let usrewritetheraisingand loweringoperators inthefollowingway
L +
(s;n)=u(s;n)I + p
(s 1)(s) 1
rx(s) E ;
L (s;n)=v(s;n)I+ p
(s)(s+1) 1
x(s) E
+
where, asbefore,(s)=(s)+(s)x(s
2 ), and
u(s;n)=
n
[n℄
q
n (s)
0
n
(s)
rx(s) ;
v(s;n)=
n
[n℄
q
n (s)
0
n +
n x(s
1
2 )+
2n
[2n℄
q
(x(s)
n )
(s)
x(s) :
Proposition 4.1 Thefuntionsu(s;n)andv(s;n)satisfy u(s+1;n)=v(s;n+1), or,
equivalentlyu(s+1;n 1)=v(s;n).
Theproofoftheabovepropositionisstraightforwardbutumbersome. Wewillinlude
it inappendixA.Ifwenowalulate
L (s;n+1)L +
(s;n)=v(s;n+1)u(s;n)+(s)(s+1)
1
x(s)
2
+
+u(s+1;n)
p
(s 1)(s) 1
rx(s)
E +
p
(s)(s+1) 1
x(s) E
+
;
and substitute thevaluesfor u(s;n),v(s;n) and H(s;n) we get
L (s;n+1)L +
(s;n)=h
(n)I+u(s+1;n)H(s;n);
where thefuntion
h
(n)=
n
[n℄
q
n (s+1)
0
n
(s+1)
rx(s+1)
n
[n℄
q
n (s)
0
n
n x(s
1
2 )
+
n
[n℄
q
n (s+1)
0
n
(s)
x(s) ;
is independent of s. In fat, applying the last equality to the orthonormal funtion
'
n
(s) and takinginto aount (28) and (29),
h
(n)=
2n
[2n℄
q
2n+2
[2n+2℄
q
n
n+1 :
Similarly,
L +
(s;n 1)L (s;n)=h
(n)I+u(s;n 1)H(s;n);
where
h
(n)=
n
[n℄
q
n (s 1)
0
n +
2n
[2n℄
q
(x(s 1)
n )+
n x(s
3
2 )
n
[n℄
q
n (s)
0
n +
2n
[2n℄
q
(x(s)
n )+
(s)
rx(s)
n
[n℄
q
n (s)
0
n +
2n
[2n℄
q
(x(s)
n )
(s 1)
x(s 1)
;
is independent of s. Furthermore,applyingthelastexpressionto thefuntions '
n (s),
and takinginto aount (28) and (29), we obtain
h
(n)=
2n 2
[2n 2℄
q
2n
[2n℄
q
n 1
n :
Remark: Notiethat h
(n+1)=h
(n).
equation for orthonormal funtions '
n
(s), admits the following fatorization {usually
alled the Infeld-Hull-typefatorization{
u(s+1;n)H(s;n)=L (s;n+1)L +
(s;n) h
(n)I; (32)
and
u(s;n)H(s;n+1)=L +
(s;n)L (s;n+1) h
(n)I; (33)
respetively.
Remark: Substituting in the above formulas the expression x(s) = s we obtain the
orrespondingresultsfortheuniformlattieases(Hahn,Kravhuk,Meixnerand
Char-lier),onsideredbeforebyseveralauthors,seee.g. [6 ,18 ,27 ℄andbytakingappropriate
limits(see e.g. [17 , 23 ℄),we anreoverthelassialontinuousase(Jaobi,Laguerre
and Hermite).
5 Appliations to some q-normalized ortogonal funtions
For the sake of ompleteness we will apply the above results to several families of
orthogonal q-polynomials and theirorresponding orthonormal q-funtionsthatare of
interest and appear in several branhes of mathematial physis. They are the
Al-Salam &Carlitz polynomials I and II, the bigq-Jaobi polynomials,the dual q-Hahn
polynomials,theontinuousq-Hermiteandtheelebratedq-Askey{Wilsonpolynomials.
Themaindataforthese polynomialsaretakenfromtheniesurvey[17 ℄exeptthe
ase of dualq-Hahnpolynomials[3℄. Nevertheless,they an beobtained alsofrom the
general formulas given inSetion2.
Finally,let uspoint outthatsimilarfatorization formulas wereobtained byother
authors, e.g. Millerin[21 ℄ onsideredthepolynomialsonthelinearexponentiallattie
and Bangerezako studied the Askey{Wilson ase. Our main aim inthis setion is to
show how our general formulas lead, in a very easy way, to the needed fatorization
formulas ofseveral familiesfornormalizedfuntions {notpolynomials{.
5.1 The Al-Salam & Carlitz funtions I and II
TheAl-Salam&CarlitzpolynomialsI(andII)appearinertainmodelsofq-harmoni
osillator , see e.g. [5 , 8 , 9 , 22 ℄. They are polynomials on the exponential lattie
x(s)=q s
x, dened[17 ℄ by
U (a)
n
(x;q)=( a) n
q
n
2
2
1 q
n
;x 1
q; qx
a
0
!
;
and onstitute anorthogonal familywiththe orthogonality relation(15)
Z
1
a U
(a)
n
(x;q)U (a)
m
(x;q)!(x)d
q x=d
2
n Æ
nm ;
where
!(x)=(qx;a 1
qx;q)
1
; and d 2
n
=( a) n
(1 q)(q;q)
n (q;a;a
1
q;q)
1 q
n
2
:
Asusual, (a
1 ;;a
p ;q)
n =(a
1 ;q)
n (a
p ;q)
n
,and (a;q)
1 =
Q
1
k=0
(1 aq k
(x)=(x 1)(x a); (x)=e(x)= 0
x+(0); being 0
= q
1=2
1 q
; (0)=q 1=2
1+a
q 1 :
The eigenvalues
n
and theoeÆients ofthe TTRRaregiven by
n =[n℄
q q
1 n=2
q 1
and
n
=1;
n
=(1+a)q n
;
n =aq
n 1
(q n
1);
respetively. Inthisasewe have
e 00
=1; e 0
(0)=
a+1
2
; e(0)=a; 0
n =
q 1
2 n
1 q ;
n (0)=q
1 n
2 a+1
q 1 :
The orrespondingnormalized funtions(25) are
'
n (x)=
v
u
u
t(qx;a 1
qx;q)
1 ( a)
n
q
n
2
(1 q)(q;q)
n
(q;a;q=a;q)
1 2
'
1 q
n
;x 1
q; qx
a
0
!
:
Deningnow theHamiltonianforthese funtions'
n (x)
H(x;n)= p
a(x 1)(x a)
x(1 q 1
)
E + p
a(qx 1)(qx a)
x(q 1) E
+
+
q 1 n
1 q x+
q(a+1)
q 1 [2℄
q
k
q x
1
I;
and usingthatu(x;n)= aq
1 q x
1
,v(x;n)=u(qx;n 1)= a
1 q x
1
,thus
L +
(x;n)=u(x;n)I+q p
a(x 1)(x a)
x(q 1)
E ; where E f(x)=f(q 1
x);
and
L (x;n)=v(x;n)I+ p
a(qx 1)(qx a)
x(q 1)
E +
; where E +
f(x)=f(qx);
wehave
L (x;n+1)L +
(x;n)= aq
1 n
(q n+1
1)
(q 1) 2
I+v(x;n+1)H(x;n);
and
L +
(x;n 1)L (x;n)= aq
2 n
(q n
1)
(q 1) 2
I+u(x;n 1)H(x;n);
whihgivethefatorizationformulasfortheAl-Salam&CarlitzfuntionsI. Ifwenow
taking into aount that (see [17 , p. 115℄)
V (a)
n
(x;q)=U (a)
n (x;q
1
);
then, thefatorization forthe Al-Salam&Carlitz funtionsII
'
n (s)=q
s
2
v
u
u
t a
s+n
(aq;q)
1 q
n+1
2
(q;aq;q)
s
(1 q)(q;q)
n 2
0 q
n
;x
q; q
n
a !
;
followsfromthefatorizationfortheAl-Salam&CarlitzfuntionsIsimplybyhanging
q to q 1
Now we will onsider the most general family of q-polynomials on the exponential
lattie,theso-alledbigq-Jaobipolynomials,thatappearintherepresentationtheory
of the quantum algebras [31 ℄. TheywereintroduedbyHahn in1949 and are dened
[17 ℄ by
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
q;q
aq;q
!
; x(s)=q s
x:
They onstitutean orthogonal family,i.e.,
Z
aq
q
!(x)P
n
(x;a;b;;q)P
n
(x;a;b;;q)d
q x=d
2
n Æ
nm
where
!(x)= (a
1
x;q)
1 (
1
x;q)
1
(x;q)
1 (b
1
x;q)
1 ;
d 2
n =
aq(1 q)(q;=a;aq=;abq 2
;q)
1
(aq;bq;q;abq=;q)
1
(1 abq)(q;bq;abq=;q)
n ( a)
n
q
n
2
(abq;abq n+1
;abq n+1
)
n
:
Theysatisfythediereneequation (1) with
(x)=q 1
(x aq)(x q) and (x)=e(x)= 0
x+(0);
where
0
=
1 abq 2
(1 q)q 1=2
and (0)=q 1=2
a(bq 1)+(aq 1)
1 q
;
and
n = q
n=2
[n℄
q
1 abq n+1
1 q :
They satisfyaTTRR, whoseoeÆientsare
n
=1;
n
=1 A
n C
n
;
n =C
n A
n 1 ;
where
A
n =
(1 aq n+1
)(1 q n+1
)(1 abq n+1
)
(1 abq 2n+1
)(1 abq 2n+2
) ;
C
n
= aq n+1
(1 q n
)(1 bq n
)(1 ab 1
q n
)
(1 abq 2n
)(1 abq 2n+1
) :
Also, we have
e 00
=
1+abq 2
q
; e 0
(0)=
abq+aq+a+
2
; (0)e =aq;
0
n =
q n
abq n+2
q 1=2
(1 q) ;
n (0)=q
1 n
2 a(bq
1+n
1)+(aq 1+n
1)
1 q
:
The normalized bigq-Jaobifuntions aredenedby
'
n (s)=
s
(x=a;x=;q)
1
(aq;bq;abq=;q)
1
(abq;aq;aq;q;q;q)
n ( a)
n
(x;bx=;=a;aq=;abq 2
;q)
1
(1 q)aq(1 abq)(q;bq;abq=;q)
n
3
2 q
n
;abq n+1
;x
q;q
aq;q
!
H(x;n)= p
a(x q)(x aq)(x q)(bx q)
x(q 1)
E +q p
a(x 1)(x a)(x )(bx )
x(q 1)
E +
+
1+abq 2n+1
q n
(1 q) x
q(a+ab++a)
1 q
+
aq(q+1)
1 q x
1
I:
Furthermore,
u(x;n)= abq
n+1
1 q
x+D
n aq
2
q 1 x
1
and v(x;n)= abq
n+1
1 q
x+D
n 1
aq
q 1 x
1
;
where
D
n =
ab(ab+a+a+)q 2n+3
a(b++ab+b)q n+2
(1 abq 2n+2
)(1 q)
;
thus
L +
(x;n)=u(x;n)I+ p
a(x q)(x aq)(x q)(bx q)
x(q 1)
E ; where E f(x)=f(q 1
x);
and
L (x;n)=v(x;n)I +q p
a(x 1)(x a)(x )(bx )
x(q 1)
E +
; where E +
f(x)=f(qx);
so
L (x;n+1)L +
(x;n)=Æ
n+1
n+1
I +v(x;n+1)H(x;n);
L +
(x;n 1)L (x;n)=Æ
n
n
I+u(x;n 1)H(x;n);
where
Æ
n =
(1 abq 2n 1
)(1 abq 2n+1
)
q 2n 1
(q 1) 2
:
The above formulas are the fatorization formulas for the family of the big q-Jaobi
normalized funtions.
Sine all disrete q-polynomials on the exponential lattie x(s) =
1 q
s
+
3 |the
so alled, q-Hahn lass| an be obtained from the big q-Jaobi polynomials by a
ertain limit proess (see e.g. [2, 17 ℄, then from the above formulas we an obtain
the fatorization formulas for the all other ases in the q-Hahn tableau. Of speial
interest are the q-Hahn polynomials and the big q-Laguerre polynomials, whih are
partiular ases of the big q-Jaobi polynomials when = q N 1
, N = 1;2;:::, and
=0,respetively.
5.3 The q-dual-Hahn funtions
In this setion we will deal with the q-dual-Hahn polynomials, introdued in [3 , 24 ℄
and losely related with the Clebsh-Gordon oeÆients of the q-algebras SU
q
(2) and
SU
q
(1;1) [3℄. Theyare denedon thelattie x(s)=[s℄
q [s+1℄
q by
W
n
(x(s);a;b)
q =
( 1) n
(q a b+1
;q)
n (q
a++1
;q)
n
q
n=2(3a b++1+n)
k n
q (q;q)
n 3
2 q
n
;q a s
;q a+s+1
q;q
q a b+1
;q a++1
!
(s)= q 1 2 ((b 1) 2 (2s 1)(a+)) (1 q) 2(a+ b)+1 (q s a+1 ;q s +1 ;q s+b+1 ;q b s ;q) 1
(q;q;q s+a+1 ;q s++1 ;q) 1 ; where 1 2
a<b 1; jj<a+1;and forthisweight funtionthenorm is
d 2 n = q 1 4
( 4ab 4b+6a+6 8b+6+4n(a+ 2b) n 2 +17n+2b 2 ) (1 q) 2(a+ b+1)+3n (q b n
;q b a n
;q) 1 [n℄ q !(q;q a++n+1 ;q) 1 :
These polynomials satisfyaTTRR (17)with
n =1; n =q 1 2
(2n b++1)
[b a n+1℄
q [a++n+1℄ q +q 1 2
(2n+2a+b+1)
[n℄
q
[b n℄
q +[a℄ q [a+1℄ q ; n =q 2n++a b
[a++n℄
q
[b a n℄
q
[b n℄
q [n℄
q ;
and theseondorder diereneequation(1),whoseeigenvaluesare
n =[n℄ q q 1 2 n 2 and
(s)=q 1
2
(s++a b+2)
[s a℄
q [s+b℄
q [s ℄
q
and (x)=e(x)= 0
x+(0);
with 0
= 1 and (0)=q 1
2
(a b++1)
[a+1℄
q
[b 1℄
q +q 1 2 ( b+1) [b℄ q [℄ q :
Also we willneedthevalues
e 00 = k q ; e 0 (0)= 1 2k q (2[2℄ q q 1 2 b q 1 2 +a q 3 2 +a+ b q 1 2 + ); e (0)= 1 2k 3 q (2q
1+ab
+q 1
+q+2q 1+ b
+2q 1+a+
(1+q)(q b +q a +q +q 1+a+b
)); 0 n = q n ; n (0)=q 1 2
( b n+1)
[+ n 2 ℄ q [b n 2 ℄ q +q 1 2 (a+b+1
n 2 ) [a+ n 2 +1℄ q
[b n 1℄
q ;
In this ase, the Hamiltonian, assoiated with the q-dual Hahn normalized funtions
p (s)=d 2 n W n
(x(s);a;b)
q ,is
H(s;n)=q 1
2
(+a b+2) q
([s+1℄ 2 q [a℄ 2 q )([b℄ 2 q
[s+1℄ 2
q
)([s+1℄ 2 q [℄ 2 q )
[2s+2℄
q E + + q 1 2 (+a b+2) q ([s℄ 2 q [a℄ 2 q )([b℄ 2 q [s℄ 2 q )([s℄ 2 q [℄ 2 q ) [2s℄ q E q 1 2 n 2 [n℄ q
[2s+1℄
q I+ q 1 2 (+a b+2) [s a℄ q [s+b℄
q
+[s ℄
q
[2s℄
q
[s+1 a℄
q
[s+1+b℄
q
[s+1 ℄
q
[2s+2℄
q
I;
where E +
f(s)=f(s+1) and E f(s)=f(s 1). Then,usingthat
u(s;n)=q 1
2 n
2
x(s+n=2) q 1 2 + n 2 (q 1 2
( b n+1)
[+ n 2 ℄ q [b n 2 ℄ q + q 1 2 (a+ b+1 n 2 ) [a+ n 2 +1℄ q
[b n 1℄
q ) q 1 2 (s++a b+2) [s a℄ q [s+b℄
q [s ℄ q [2s℄ q ;
and taking into aount that v(s;n)=u(s+1;n 1), we nd
L +
L (s;n)=v(s;n)I+q 1
2
(+a b+2) q
([s+1℄ 2
q [a℄
2
q )([b℄
2
q
[s+1℄ 2
q
)([s+1℄ 2
q [℄
2
q )
[2s+2℄
q
E +
:
Thus
L (s;n+1)L +
(s;n)=q 2n
n+1
I+v(s;n+1)H(s;n);
and
L +
(s;n 1)L (s;n)=q 2n+2
n
I +u(s;n 1)H(s;n);
are thefatorization formulas forthe q-dualHahnnormalized funtions.
5.4 The Askey{Wilson funtions
FinallywewillonsiderthefamilyofAskey{Wilsonpolynomials. Theyarepolynomials
on thelattie x(s)= 1
2 (q
s
+q s
)x,dened by[17℄
p
n
(x(s);a;b;;d)= (ab;q)
n (a;q)
n (ad;q)
n
a n
4
3 q
n
;q n 1
abd;ae i
;ae i
q;q
ab;a;ad
!
;
i.e., they orrespond to the general ase (23) when q s1
= a, q s2
=b, q s3
=,q s4
=d.
Their orthogonalityrelationisof theform
Z
1
1 !(x)p
n
(x;a;b;;d)p
m
(x;a;b;;d) p
1 x 2
q dx=Æ
nm d
2
n
; q
s
=e i
; x=os;
where
!(x)=
h(x;1)h(x; 1)h(x;q 1
2
)h(x; q 1
2
)
2
q (1 x
2
)h(x;a)h(x;b)h(x;)h(x;d)
; h(x;)= 1
Y
k=0
[1 2xq k
+ 2
q 2k
℄;
and thenorm isgiven by
d 2
n =
(abdq n 1
;q)
n (abdq
2n
;q)
1
(q n+1
;abq n
;aq n
;adq n
;bq n
;bdq n
;dq n
;q)
1 :
The Askey{Wilsonpolynomialssatisfythediereneequation (1) with
(s)= q
2s+1=2
2
q (q
s
a)(q s
b)(q s
)(q s
d);
q =(q
1
2
q 1
2
)
and (x)=e(x)= 0
x+(0), where
0
=4(q 1)(1 abd); (0)=2(1 q)(a+b++d ab abd ad bd):
Furthermore, theysatisfy theTTRR(17) withoeÆients
n
=1;
n =
a+a 1
(A
n +C
n )
2
;
n =
C
n A
n 1
4 ;
where A
n ,C
n
aredened by
A
n =
(1 abq n
)(1 aq n
)(1 adq n
)(1 abdq n 1
)
a(1 abdq 2n 1
)(1 abdq 2n
)
;
C
n =
a(1 q n
)(1 bq n 1
)(1 bdq n 1
)(1 dq n 1
)
(1 abdq 2n 2
)(1 abdq 2n 1
)
and whoseeigenvaluesare
n
=4q (1 q )(1 abdq ). Inaddition,wehave
e 00
= 4(q 1) 2
(1+abd)q 1=2
;
e 0
(0)=(q 1) 2
(a+b++d+ab+abd+ad+bd)q 1=2
;
e
(0)=(q 1) 2
(1 ab a ad b bd d+abd)q 1=2
;
0
n =4q
n
(q 1)(1 abdq 2n
);
n
(0)=2(q 1)( a b d+(ab+abd+ad+bd)q n
)q n=2
:
Dening now the normalized funtions (see (15)) p
!(x)=d 2
n p
n
(x;a;b;;d), the
orre-sponding HamiltonianH(s;n) is
H(s;n)= 2q
3=2
[2s 1℄
q
G(s;a;b;;d)E + 2q
3=2
[2s+1℄
q
G(s+1;a;b;;d)E +
+
2 q
2s+1=2 Q
4
i=1 (1 q
s
i +s
)
[2s+1℄
q
+q
2s+1=2 Q
4
i=1 (q
s
q s
i
)
[2s 1℄
q +
q n+1
2
q (1 q
n
)(1 abdq n 1
)[2s℄
q !
I
where
G(s;a;b;;d)= v
u
u
t 4
Y
i=1 (1 2q
s
i
q 1=2
x(s 1=2)+q 1
q 2s
i
) ;
Wenow dene
u(s;n)=D
n x
n
(s)+D
n E
n +q
2s+1=2 (q
s
a)(q s
b)(q s
) (q s
d)
[2s 1℄
q
where
D
n = 4q
n=2+1=2
(q 1)(1 abdq n 1
):
E
n =
( a b d+(ab+abd+ad+bd)q n
)q n=2
2(1 abdq 2n
)
:
Takinginto aount that v(s;n)=u(s+1;n 1), we nd
L +
(s;n)=u(s;n)I + 2q
3=2
[2s 1℄
q
G(s;a;b;;d)E ;
L (s;n)=v(s;n)I+ 2q
3=2
[2s+1℄
q
G(s+1;a;b;;d)E +
;
where E f(s)=f(s 1) and E +
f(s)=f(s+1). Thus,
L (s;n+1)L +
(s;n)=D
2n D
2n+2
n+1
I+v(s;n+1)H(s;n);
and
L +
(s;n 1)L (s;n)=D
2n 2 D
2n
n
I+u(s;n 1)H(s;n);
when a=b==d=0,i.e., theontinuous q-Hermitepolynomials
H
n
(xjq)=2 n
e in
2
0 q
n
;0
q;q n
e 2i
|
!
; x=os:
These polynomialsare loselyrelated with theq-harmoni osilatormodelintrodued
byBiedenharn[13 ℄ and Mafarlane[19 ℄, asitwaspointedoutin[9 ℄, wherethe
fator-izationfortheontinuousq-Hermitepolynomialswereonsideredrst. Ifwesubstitute
a=b==d =0intheabove formulas,we obtainthefatorization fortheq-Hermite
funtions
'
n (x)=
s
h(x;1)h(x; 1)h(x;q 1=2
)h(x; q 1=2
)(q n+1
;q)
1
2
q (1 x
2
)
H
n (xjq):
In fat,sine forontinuous q-Hermite polynomials
(s)= 2
q q
2s+1=2
; (s)=4(q 1)x(s);
n =4q
n+1
(1 q n
);
and the oeÆients for the three-term reurrene relation are
n
= 1,
n
= 0,
n =
(1 q n
)=4, thenweobtain
H(s;n)= 2q
3=2
[2s 1℄
q E +
2q 3=2
[2s+1℄
q E
+
+2 q
2s+1=2
[2s+1℄
q +
q 2s+1=2
[2s 1℄
q q
n+1
2
q (1 q
n
)[2s℄
q !
I;
L +
(s;n)= 4q
n=2+1=2
(q 1)x(s+n=2)+ q
2s+1=2
[2s 1℄
q !
I+ 2q
3=2
[2s 1℄
q E ;
L (s;n)= 4q n=2+1
(q 1)x(s+n=2+1=2)+ q
2s+5=2
[2s+1℄
q !
I+ 2q
3=2
[2s+1℄
q E
and h
(n)=4 2
q q
2n+1
(1 q n
).
Aknowledgements
The authorsthank N. Atakishiyev and Yu. F. Smirnovforinterestingdisussionsand
remarks thatallowedusto improvethispapersubstantially,aswellasthereferees for
theirremarks. The work hasbeenpartiallysupported bythe MinisteriodeCienias y
Tenologa of Spain under the grant BFM-2000-0206-C04 -02 , theJunta de Andalua
under grant FQM-262 and theEuropean proyetINTAS-2000-272.
Appendix A
Here,forthesakeofompleteness,wewillproveProposition4.1,byshowingthatu(s+1;n)
v(s;n+1)=0. Todothat,westartwithomputingthedierene
u(s+1;n) v(s;n+1)=
n
[n℄
q
n (s+1)
0
n
(s)
x(s) +
n+1
[n+1℄
q
n+1 (s)
0
n+1
n+1 x s
1
2
2n+2
[2n+2℄
q
(x(s)
n+1 )+
(s)x s 1
2
Nowweusetheexpansion
n
(s+1)=
n x
n
(s+1)+
n (0). Sine (x 2 (s)) x(s) = x 2
(s+1) x 2
(s)
x(s+1) x(s)
=x(s+1)+x(s)=C
1 q
s
(q+1)+C
2 q
s
(q 1
+1)+2C
3 = (C 1 q s+ 1 2 +C 2 q s 1 2 )[2℄ q +2C 3 =[2℄ q x 1
(s)+(2 [2℄
q )C 3 ; x(s)x(s 1 2
)=x(s)(C
1 q
s 1
2
(q 1)+C
2 q s+ 1 2 (q 1
1))=x(s)(C
1 q s C 2 q s )k q = (C 2 1 q 2s C 2 2 q 2s )k q +C 3 (C 1 q s C 2 q s )k q ; wherek q =q 1 2 q 1 2 , x(s) x(s)x(s 1 2 ) = (C 2 1 q 2s+1 +C 2 2 q 2s 1 )[2℄ q +C 3 (C 1 q s+ 1 2 +C 2 q s 1 2 ) C 1 q s+ 1 2 C 2 q s 1 2 ! k q ; and x(s) x(s 1 2 ) = x(s) (C 1 q s C 2 q s )k q = C 1 q s+ 1 2 +C 2 q s 1 2 C 1 q s+ 1 2 C 2 q s 1 2 k q : Then (s) x(s) = x(s) e (s) 1 2 e (s)x(s 1 2 ) = = x(s) e 00 2 x 2
(s)+e 0
(0)x(s)+e(0) 1
2 (
0
x(s)+(0))x(s 1 2 ) = e 00 2 ( [2℄ q x 1
(s)+(2 [2℄
q )C
3 )+e
0 (0) 1 2 (0) C 1 q s+ 1 2 +C 2 q s 1 2 C 1 q s+ 1 2 C 2 q s 1 2 ! k q 1 2 0 [2℄ q (C 2 1 q 2s+1 +C 2 2 q 2s 1
)+C
3 (C 1 q s+ 1 2 +C 2 q s 1 2 ) C 1 q s+ 1 2 C 2 q s 1 2 ! k q :
Thisyieldsforu(s+1;n) v(s;n+1)theexpression
= n [n℄ q x n
(s+1)+ n [n℄ q n (0) 0 n e 00 2 [2℄ q x 1 (s)+ C 3 2 (2 [2℄ q )e 00 +e 0 (0) e 0 2 [2℄ q (C 2 1 q 2s+1 +C 2 2 q 2s 1 ) C 1 q s+ 1 2 C 2 q s 1 2 + C 3 x 1 (s) C 2 3 C 1 q s+ 1 2 C 2 q s 1 2 k q (0) 2 x 1 (s) C 3 C 1 q s+ 1 2 C 2 q s 1 2 k q + n+1
[n+1℄
q n+1 (s) 0 n+1 n+1 x s 1 2 2n+2
[2n+2℄
q C 1 q s +C 2 q s +C 3
[n+1℄
q n (0) 0 n +
[n+2℄
q n+1 (0) 0 n+1 C 3
(1+[n+1℄
q
[n+2℄
q ) + (s)x s 1 2 x(s) :
Next,weexpandx
n (s)and e 00 2 [2℄x 1
(s),makesomestraightforwardalulationsand usethe
identities: n [n℄ q n (0) 0 n
+[n+1℄
q
2n+2
[2n+2℄
q n (0) 0 n = n [n℄ q
+[n+1℄
q
2n+2
[2n+2℄
q n (0) 0 n
= [n+2℄
q n (0); n+1
[n+1℄
q n+1 (s) 0 n+1
[n+2℄
q
2n+2
[2n+2℄
q n+1 (0) 0 n+1
=[n+1℄
q n+1 (0)+ n+1
[n+1℄
q x
n
[n℄
q (C
1 q
s+1+ n
2
+C
2 q
s 1 n
2
) e 00
2 [2℄
q (C
1 q
s+ 1
2
+C
2 q
s 1
2
)
2n+2
[2n+2℄
q (C
1 q
s
+C
2 q
s
)
n+1 (C
1 q
s
C
2 q
s
)k
q +
1
2
0
(C
1 q
s+ 1
2
+C
2 q
s 1
2
)(q+q 1
)= C
1 q
s
0
2 (q
n+ 1
2
+q 1
2
)+
C
1 q
s
e 00
2(q 1
2
q 1
2
) (q
n+ 1
2
q 1
2
)+ C
2 q
s
0
2 (q
n 1
2
+q 1
2
)+ C
2 q
s
e 00
2(q 1
2
q 1
2
) ( q
n 1
2
+q 1
2
)=
=
n+1
[n+1℄
q (C
1 q
s+ n+1
2
+C
2 q
s n+1
2
);
wend
=
n+1
[n+1℄
q C
1 q
s+ n+1
2
+C
2 q
s n+1
2
+C
3
n
[n℄
q
[n+2℄
q
n (0) C
3 e 00
e 0
(0)+ 1
2
0
C
3 k
q +
1
2 (0)k
q
+[n+1℄
q
n+1 (0)+
n+1
[n+1℄
q (C
1 q
s+ n+1
2
+C
2 q
s n+1
2
+C
3 )+
2n+2
[2n+2℄
q C
3 ([n+1℄
q
[n+2℄
q ):
Finally,wesubstitutetheexpressionfor
n
(0)andusetheidentities
[n+2℄
q [n℄
q
1+[n+1℄
q [n+1℄
q =0;
[n+2℄
q (q
n=2
+q n=2
)+k
q
+[n+1℄
q (q
(n+1)=2
+q (n+1)=2
)=0;
andtheresultfollows.
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