On the factorization of the difference equation of hypergeometric-type in nonuniform lattices.

Texto completo

(1)

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 .

(2)

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

(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

(4)

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

(5)

(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:

(6)

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

(7)

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

(8)

= 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

+

(9)

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).

(10)

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

(11)

(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

(12)

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

!

(13)

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

!

(14)

(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 +

(15)

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

)

(16)

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);

(17)

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

(18)

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

(19)

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.

Referenes

[1℄ R.

Alvarez-NodarseandJ.Arvesu,Ontheq-polynomialsontheexponentiallattiex(s)=

1 q

s

+

3

.Integral Trans.andSpeial Funt.8(1999),299-324.

[2℄ R.

Alvarez-Nodarse y J. C. Medem, q Classial polynomials and the q Askey and

Nikiforov-Uvarov.J.Comput. Appl.Math.(2001),inpress.

[3℄ R.

Alvarez-Nodarse and Yu.F. Smirnov, q-Dual Hahn polynomials on the non-uniform

lattiex(s)=[s℄

q [s+1℄

q

andtheq-algebrasSU

q

(1;1)andSU

q

(2).J.Phys. A.29(1996),

1435-1451.

[4℄ N. M. Atakishiyev, Constrution of the dynamial symmetry group of the relativisti

harmoniosillatorbytheInfeld-Hullfatorizationmethod.InGroupTheoretialMethods

inPhysis,M.SerdarogluandE.Inonu.(Eds.).LetureNotesinPhysis,Springer-Verlag,

180,1983,393-396.Ibid.Theor. Math. Phys. 56(1984),563-572.

[5℄ R.AskeyandS. K.Suslov,Theq-harmoniosillatorandtheAl-SalamandCarlitz

poly-nomials.Lett.Math. Phys.29 (1993),123{132.

[6℄ N.M.AtakishiyevandB.Wolf,ApproximationonanitesetofpointsthroughKravhuk

funtions.Rev.Mex.Fis.40 (1994),1055-1062.

[7℄ N.M.Atakishiyev,M.Rahman,andS.K.Suslov, OnClassialOrthogonalPolynomials.

Construtive Approx. 11(1995),181-226.

[8℄ N.M.AtakishiyevandS.K.Suslov,Diereneanalogsoftheharmoniosillator.Theoret.

andMath.Phys. 85(1991),442-444.

[9℄ N. M.Atakishiyevand S. K.Suslov, A realizationof the q-harmoniosillator.Theoret.

(20)

bergq-algebra.J. Math. Phys.35 (1994),3253-3260.

[11℄ G.Bangerezako,Disrete Darboux transformationfor disretepolynomialsof

hypergeo-metritype.J. PhysA: Math.Gen.31 (1998),2191-2196.

[12℄ G.Bangerezako,ThefatorizationmethodfortheAskey{Wilsonpolynomials.J.Comput.

Appl.Math.107(1999),219-232.

[13℄ L.C.Biedenharn, ThequantumgroupSU

q

(2) andaq-analogueof thebosonoperators.

J. Phys. A.22 (1989),L873-L878.

[14℄ G.Darboux,Theorie desSurfaes.volII (Paris,Gauthier-Villars).

[15℄ G. Gasper and M. Rahman, Basi Hypergeometri Series. Cambridge University Press,

1990.

[16℄ L.InfeldandT.E.Hull,Thefatorizationmethod.Rev.ModernPhysis23(1951),21{68.

[17℄ R.KoekoekandR.F.Swarttouw,TheAskey-shemeofhypergeometriorthogonal

polyno-mialsanditsq-analogue.ReportsoftheFaultyofTehnialMathematisandInformatis

No.98-17. DelftUniversityofTehnology,Delft,1998.

[18℄ M. Lorente, Raising and lowering operators, fatorization method and

dieren-tial/diereneoperatorsofhypergeometritype.Preprint2000.

[19℄ A. J. Mafarlane,On q-analoguesof the quantum harmoniosillator and the quantum

groupSU

q

(2).J. Phys. A.22(1989),4581-4588.

[20℄ W. Miller (Jr.), Lie theory and diereneequations. I. J. Math. Anal. Appl. 28 (1969),

383{399.

[21℄ W. Miller (Jr.), Lie theory and q-dierene equations. SIAM J. Math. Anal. 1 (1970),

171{188.

[22℄ Sh. M. Nagiyev, Dierene Shrodinger equation and q-osillator model, Theoret. and

Math.Phys. 102(1995),180-187.

[23℄ A. F. Nikiforov,S. K. Suslov,and V. B. Uvarov, Classial Orthogonal Polynomials of

a Disrete Variable. SpringerSeries in Computational Physis.Springer-Verlag, Berlin,

1991.

[24℄ A.F.NikiforovandV.B.Uvarov,PolynomialSolutionsofhypergeometritypedierene

equations and their lassiation. Integral Transforms and Speial Funtions. 1 (1993),

223-249.

[25℄ E.Shrodinger,Amethodofdeterminingquantum-mehanialeigenvaluesand

eigenfun-tions.Pro. Roy. Irish.Aad.46A(1940),9-16.

[26℄ E.Shrodinger,Thefatorizationofthehypergeometriequation.Pro. Roy.Irish. Aad.

47A(1941),53-54.

[27℄ Yu. F. Smirnov, On fatorization and algebraization of dierene equation of

hyperge-ometri type. In Pro. Internat. Workshop on Orthogonal Polynomials in Mathematial

Physis, M. Alfaroet al.Eds.Serviio dePubliaiones dela UniversidadCarlos III de

Madrid,(1997)153-161.

[28℄ Yu.F.Smirnov,Finitediereneequationsandfatorizationmethod.InPro.5thWigner

Symposium,P.KasperkovitzandD.Grau Eds.WorldSienti(1998)148-150.

(21)

exponentialdisretespetra.Lett. Math.Phys. 29(1993),63{73.

[31℄ N. Ja. Vilenkinand A.U.Klimyk,Representationsof Lie Groups andSpeial Funtions.

Figure

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Referencias

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