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(1)

x(s) =

1 q

s

+

3 .

R.

Alvarez-Nodarse

Departamento de AnalisisMatematio Universidadde Sevilla

Apdo. 1160, E-41080 Sevilla, Spain

and Instituto Carlos I de Fsia Teoria y Computaional

Universidad de Granada E-18071,Granada, Spain

and

J.Arves u y

Departamento de Matematias

E.P. S.UniversidadCarlos III de Madrid

Butarque 15,28911, Leganes, Madrid,Spain

Otober 13,2003

Dediated toVasily B. Uvarov(19/10/1929 { 23/10/1997)

Keywordsandphrases: disretepolynomials,q-polynomials,basihypergeometri

series, non-uniformlatties, q-Charlierpolynomials.

AMS (MOS,1995) subjet lassifiation: 33D25

Abstrat

Themaingoalofthepresentpaperistoontinuethestudyoftheq-polynomials

onnon-uniformlattiesbyusingtheapproahintroduedbyNikiforovandUvarov

in 1983. We onsider the q-polynomials on the non-uniform exponential lattie

x(s)=

1 q

s

+

3

andstudysomeoftheirproperties(dierentiationformulas,

stru-turerelations,representationintermsofhypergeometriandbasihypergeometri

funtions,et). Speialemphasisisgiventoaq-analogueoftheCharlierorthogonal

polynomials. Forthesepolynomials(Charlier)weomputethemaindata,i.e.,the

oeÆientsofthethree-term reurrenerelation, struturerelation, thesquare of

thenorm,et,in theexponentiallattiesx(s)=q s

andx(s)= q

s

1

q 1

,respetively.

IntegralTransform. Speial Funt. 8(1999)299-324

E-mail: ranes.es, Fax: (+34)95-455-79-72

y

(2)

Inthelastyears thestudyofthedisreteanaloguesofthelassialspeialfuntions

and, inpartiular,theorthogonal polynomials,hasreeived an inreasinginterest (for

a review see [20, 25 , 26 , 34 ℄). Speial emphasis was given to the q-analogues of the

orthogonal polynomials orq-polynomials,whihare loselyrelated with dierent

top-is in other elds of atual siene: Mathematis (e.g., ontinued frations, eulerian

series, theta funtions,ellipti funtions,...; see forinstane [5 , 24 ℄) and Physis (e.g.,

q-Shrodingerequationand q-harmoniosillators[8 , 9,10 ,15 ,16 ,18 ,30 ℄). Moreover,

theonnetionbetweentherepresentationtheoryofquantumalgebras(Clebsh-Gordan

oeÆients, 3j and 6j symbols) and the q-orthogonal polynomials is well known, (see

[4 , 27 ,28 , 31 ,39 ,41 ℄ among others.)

There exist several approahes to the study of these objets. The more standard

one is based on the fatthat these q-polynomials arespeial ases of the basi

hyper-geometri series[25℄(seealso [11 ,26 ,28 ℄). Otherapproahesarethegroup-theoretial

approah [23 , 41 ℄, the algebrai approah [33 ℄, and the dierene-equation in

non-uniform latties approah [12 , 13 , 14 , 17 , 22 , 34 , 35 , 36, 37 , 40 ℄. The distribution of

zeros of these polynomialshasbeenonsideredin[2 ℄ and[21 ℄.

In most of the papers related to the last approah (Nikiforov et al.) the authors

didn'tstudyonretefamilies(upto thepapers[19,39 ℄),althoughtheyareveryuseful

for appliations. In the present paper we will study some q-polynomials on the

ex-ponential lattie x(s) = q s

, onneting the results of dierent authors [19 ℄ (Meixner

and Kravhuk), [39 ℄ (Hahn), and the general method desribed in [34 , 37 ℄. In

par-tiular, we willderive from thegeneral formulas,obtained in[37 ℄, thehypergeometri

representationofthesepolynomialswhihhasnotbeenonsideredin[19 ℄and[39 ℄.

Par-tiular emphasis will be given to a q-analogue of theCharlier polynomials, for whih

wewillndalltheharateristiformulas,inludingnorm,three-term reurrene

rela-tion, struture relations, dierentiation formulas, representationin terms of the basi

hypergeometriseriesand soon.

It is important to remark that the above lattie x(s) = q s

is a \bad" lattie in

the sense that we an not reover the linear lattie x(s) = s by taking limits in it.

In fat, a more onvenient lattie is the lattie x(s) = q

s

1

q 1

. The eletion of suh a

lattie is determined by the fat that in the limit (when q tends to one) we reover

thelassiallinearlattie,and thenthenew polynomialsseemto be \good" analogues

of thelassial ones. In partiular all the harateristis of the resulting polynomials

willtendto thelassialones inthelimitq!1. To show thiswewillstudyindetails

theq analoguesoftheCharlierpolynomialsinbothlattiesx(s)=q s

andx(s)= q

s

1

q 1 .

The struture of the paper is as follows. In Setion 2 we summarize some of the

properties of the q-polynomials on the non-uniform latties [34 , 37℄ with speial

em-phasis intheexponentiallattie. In setion 3 we derivesome generalrelations forthe

q-polynomialsontheexponentiallattie. Inpartiular,wewillobtaintheso-alled

dif-ferentiationformulas,thestruturerelations,andanexpressioninvolvingthedierene

derivatives of the q-polynomials on the exponential lattie x(s) =

1 q

s

+

3

with the

polynomialitself. All these relations are very useful for appliations(see e.g. [3 , 29 ℄)

and theirknowledge allows to extend theresults of [6 , 38 ℄ for the orthogonal

(3)

sidered in[19 , 39℄ byusingthe approah byNikiforovand Uvarov[34 , 37 ℄. Finally,in

Setion5 we studyindetaila q-analogueof theCharlierpolynomialsinthetwo

afore-said exponentiallattiesand ndexpliitexpressions forsome oftheir harateristis.

2 General properties of the q polynomials on the

expo-nential lattie.

2.1 The hypergeometri-type dierene equation.

Letusstartwiththestudyofsome generalpropertiesoforthogonalpolynomialsof

a disretevariableon non-uniformlatties. Let

~ (x(s))

4

4x(s 1

2 )

5y(s)

5x(s) +

~ (x(s))

2

4y(s)

4x(s) +

5y(s)

5x(s)

+y(s)=0;

5f(s)=f(s) f(s 1);4f(s)=f(s+1) f(s);

(2.1)

be the seond order dierene equation of hypergeometri type for some lattie

fun-tion x(s), where 5f(s) and 4f(s),denote thebakwardand forwardnitedierene

derivatives,respetively. Here(x)~ and ~(x)arepolynomialsinx(s)ofdegree atmost

2and 1,respetively,andisaonstant. Theequation(2.1)an beobtainedfromthe

lassialdierentialhypergeometri equation

~ (x)y

00

(x)+~(x)y 0

(x)+y(s)=0;

via the disretization of the rst and seond derivatives y 0

and y 00

in an appropriate

way[34,36 ℄. Following [34 ,37℄ we willrewrite(2.1) inthe equivalentform

(s) 4

4x(s 1

2 )

5y(s)

5x(s)

+(s) 4y(s)

4x(s)

+y(s)=0;

(s)=(x(s))~ 1

2 ~

(x(s))4x(s 1

2

); (s)=~(x(s)):

(2.2)

It is known [12 , 34, 37 ℄that the above diereneequations have polynomialsolutions

of hypergeometritype ix(s) isa funtionofthe form

x(s)=

1 (q)q

s

+

2 (q)q

s

+

3

(q)=

1 (q)[q

s

+q s

℄+

3

(q); (2.3)

where

1 ,

2 ,

3 and q

=

1

2

are onstants whih, in general, dependon q. A simple

alulation shows thatthe exponentiallattie belongsto thislass. Infat wehave

lim

2 !0

x(s)= lim

!1

x(s)=

1 q

s

+

3

; (2.4)

where takes theappropriatesign aording to q>1 orq<1.

Thek-order dierene derivativeof a solutiony(s) of (2.1) ,denedby

y

k (s)

q =

4

4x

k 1 (s)

4

4x

k 2 (s)

::: 4

4x(s)

[y(s)℄4 (k)

[y(s)℄;

where

x

m

(s)=x(s+ m

2

(4)

(s)

4

4x

k (s

1

2 )

5y

k (s)

q

5x

k (s)

+

k (s)

4y

k (s)

q

4x

k (s)

+

k y

k (s)

q

=0; (2.6)

where [34 , page62, Eq. (3.1.29)℄

k (s)=

(s+k) (s)+(s+k)4x(s+k 1

2 )

4x

k 1 (s)

; (2.7)

and

k =

n +

k 1

X

m=0 4

m (s)

4x

m (s)

: (2.8)

Usually,theequations(2.2) and(2.6) are writtenintheompat form

4

4x(s 1

2 )

(s)(s) 5y(s)

5x(s)

+(s)y(s)=0; (2.9)

and

4

4x

k (s

1

2 )

(s)

k (s)

5y

k (s)

5x

k (s)

+

k

k (s)y

k

(s)=0; (2.10)

where (s)and

k

(s) arethesolutionof thePearson-type diereneequations[34℄

4

4x(s 1

2 )

[(s)(s)℄=(s)(s);

4

4x

k (s

1

2 )

[(s)

k

(s)℄=

k (s)

k (s);

(2.11)

respetively,and

k

(s) isgiven by

k

(s)=(s+k) k

Y

i=1

(s+i): (2.12)

2.2 The Rodrigues-type formula.

Itiswellknown[34 , 37℄,thatthepolynomialsolutionsofequation (2.2),denotedby

P

n (s)

q P

n (x(s))

q

, are determined,up to a normalizingfator B

n

, by the dierene

analog of theRodriguesformula[34, page 66,Eq. (3.2.19)℄

P

n (s)

q =

B

n

(s) 5

(n)

[

n

(s)℄; 5 (n)

5

5x

1 (s)

5

5x

2 (s)

::: 5

5x

n (s)

; (2.13)

where the funtion

n

(s) is given in (2.12). These polynomial solutions orrespond

to some values of

n

{the eigenvalues of equation (2.2){ and an be omputed by

substitutingthepolynomialP

n (s)

q

in(2.1)andequatingtheoeÆientsofthegreatest

powerx n

(s) (see[34, page104 ℄and [37 ℄). Then

n

= [n℄

q n

1

2 q

n 1

+q n+1

~ 0

+[n 1℄

q ~ 00

2 o

; (2.14)

where (seeEq. (2.2))

~ (s)=

~ 00

2 x(s)

2

+~ 0

(0)x(s)+(0);~ and ~(s) 0

x(s)+(0):

Here and throughoutthepaper[n℄

q

denotesthesoalledq-numbers

[n℄

q =

q n

2

q n

2

q 1

2

q 1

2 =

sinh(hn)

sinh(h)

; q =e h

2

; and

q =q

1

2

q 1

2

(5)

kn q n q

type formulais valid

y

kn (s)

q =4

(k)

P

n (s)

q =

A

nk B

n

k (s)

5 (n)

k [

n

(s)℄; (2.15)

where theoperator5 (n)

k

is denedby

5 (n)

k

f(s)= 5

5x

k+1 (s)

5

5x

k+2 (s)

5

5x

n (s)

[f(s)℄;

B

n =

4 (n)

Pn

Ann

,theeigenvalues

n

aregiven by(2.14) and

A

nk =

[n℄

q !

[n k℄

q !

k 1

Y

m=0 (

q 1

2

(n+m 1)

+q 1

2

(n+m 1)

2

!

~ 0

+[n+m 1℄

q ~ 00

2 )

: (2.16)

TheRodriges-typeformula(2.13) an be writtenalso intheform [13 , 34℄

P

n (s)

q =

B

n

(s)

Æ

Æx(s)

n

[

n (s

n

2 )℄;

Æ

Æx(s)

n

n times

z }| {

Æ

Æx(s) Æ

Æx(s)

Æ

Æx(s)

; (2.17)

where Æf(s)=5f(s+ 1

2

)f(s+ 1

2

) f(s 1

2

). Analogously,

4 (k)

P

n (s)

q =

A

nk B

n

k (s)

"

Æ

Æx(s k

2 )

#

n k

[

n (s

n

2 +

k

2

)℄: (2.18)

Sometimes, for the exponential lattie, it is better to rewrite the (2.13) in the

equivalent form [3 ℄

P

n (s)

q =

q n(n+1)

4

B

n

(s)

5

5x(s)

n

[

n (s)℄;

5

5x(s)

n

=

n-times

z }| {

5

5x(s) :::

5

5x(s)

: (2.19)

To obtainthe above formula the linearity of theoperator 5 (n)

aswell asthe identity

5x

k (s)=q

k

2

5x(s), have beenused.

2.3 Integral representation and expliit formula.

For the polynomial solutions P

n (s)

q

of the dierene equation (2.2) the following

integralrepresentationholds[12℄

P

n (s)

q =

[n℄

q !B

n

(s) 2i Z

C

n (z)x

0

n (z)

[ x

n

(z) x

n (s)℄

(n+1)

dz; (2.20)

where

[ x

k

(z) x

k (s)℄

(m)

= m 1

Y

j=0 [x

k

(z) x

k

(s j)℄; m=0;1;2::: ; (2.21)

aretheso-alledgeneralizedpowers. Inthispaperwesupposethatthefuntion

n (z)is

analytionand insidethelosedontourC oftheomplexplaneontainingthepoints

z = s;s 1;::::;s n. From the above expression we an obtainan expliitformula

for the polynomials P

n (s)

q

(6)

0;1;;n), we nd the following expression for the polynomials in the exponential

lattie x(s)=

1 q

s

+

3 ,

P

n (s)

q =

B

n q

ns+ n

4 (n+1)

n

1 (q 1)

n n

X

m=0 ( 1)

m+n

[n℄

q !q

m

2 (n 1)

[m℄

q

![n m℄

q !

n

(s n+m)

(s)

; (2.22)

or, usingthePearson-type equation (2.11), rewritteninits equivalent form

(s+1)

(s) =

(s)+(s)4x(s 1

2 )

(s+1)

; (2.23)

P

n (s)

q =

B

n q

ns+ n

4 (n+1)

n

1

(q 1) n

n

X

m=0 [n℄

q !q

m

2 (n 1)

( 1) m+n

[m℄

q

![n m℄

q !

n m 1

Y

l =0

[(s l)℄ m 1

Y

l =0

[(s+l)+(s+l)4x(s+l 1

2 )℄;

(2.24)

where it is assumed that 1

Y

l =0

f(l) 1. This is an expliitformula for thepolynomials

P

n (s)

q

interms onthepolynomials and of thediereneequation (2.2).

2.4 Hypergeometri representation.

Letusonsiderthe mostgeneralsolutionof (2.1) in thelattie x(s)=

1 q

s

+

3 . In

[37 ℄ has been proved that the most general ase in the exponential lattie is the one

when

(s)=

A(q s s

1

1)(q s s

2

1);

(s)+(s)4x(s 1

2 )=

A(q s s

1

1)(q s s

2

1):

(2.25)

In thisase, theeigenvaluesofthe orresponding diereneequation isgiven by

n =

A

2

1 q

1

2

(s1+s2+s1+s2)

[n℄

q [s

1 +s

2 s

1 s

2

+n 1℄

q

; (2.26)

andthepolynomialsolutionsof(2.1)anberepresentedasq-hypergeometrifuntions

[37 ℄

P

n (s)

q =

A

q

1 !

n

B

n q

n

2 (s

1 +s

2 n 1

2 )

(s

1 s

1 jq)

n (s

1 s

2 jq)

n

3 F

2

n;s

1 +s

2 s

1 s

2

+n 1;s

1 s

s

1 s

1 ;s

1 s

2

;q;q 1

2 (s s

2 )

!

=

(2.27)

=

A

q

1 !

n

B

n q

n

2 (s

1 +s

2 +

n 1

2 )

(s

1 s

1 jq)

n (s

2 s

1 jq)

n

3 F

2

n;s

1 +s

2 s

1 s

2

+n 1;s s

1

s

1 s

1 ;s

2 s

1

;q;q 1

2 (s s

2 )

!

;

(7)

P n (s) q = B n A 1 (q 1 2 q 1 2 ) ! n q n(n 1) 4 ns1 (q s1 s1 ;q) n (q s1 s2 ;q) n 3 ' 2 q n ;q s 1 +s 2 s 1 s 2 +n 1 ;q s 1 s q s 1 s 1 ;q s 1 s 2

;q;q s s 2 +1 ! = (2.29) = B n A 1 (q 1 2 q 1 2 ) ! n q 3n(n 1) 4 n(s 1 +s 2 s 1 ) (q s 1 s 1 ;q) n (q s 2 s 1 ;q) n3 ' 2 q n ;q

s1+s2 s1 s2+n 1

;q s s1 q s 1 s 1 ;q s 2 s 1

;q;q !

:

(2.30)

Here,the q-hypergeometrifuntionis denedby

r F p a 1 ;a 2 ;:::;a

r

b

1 ;b

2 ;:::;b

p ; q;z

= 1 X k =0 (a 1 jq) k (a 2 jq) k (a r jq) k (b 1 jq) k (b 2 jq) k (b p jq) k z k (1jq) k h k q q 1 4 k (k 1)

i p r+1 ; (2.31) where (ajq) k

isa q-analogueof thePohhammersymbol

(ajq) k = k 1 Y m=0

[a+m℄

q =

~

q (a+k)

~ q (a) ; (2.32) and ~ q

(x) istheq-analogue ofthe gammafuntionintroduedin[34 , 37 ℄. Notie that

the above denitionforthe q-hypergeometrifuntion is dierent from the one

intro-duedin[37 ℄byafator k q q 1 4 k(k 1) p r+1

,whihshouldbeinludedinorderto get

\good" limit transitions. Notie also that it oinides with the Nikiforov-Uvarov one

when r=p+1.

Theq-basi hypergeometriseriesisdenedby[25 ℄

r ' p a 1 ;a 2 ;:::;a

r

b

1 ;b

2 ;:::;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 ; (2.33) where (a;q) k = k 1 Y m=0 (1 aq m ); (2.34)

Thesetwo funtionsare relatedto eah other byformula[37 ℄

p+1 ' p q a 1 ;q a 2

;:::;q a p+1 q b 1 ;q b 2

;:::;q b

p

;q;z ! = p+1 F p a 1 ;a 2 ;:::;a

p+1

b

1 ;b

2 ;:::;b

p

;q;t !

t=t

0 ;

where t

0 =zq

1 2 P p+1 i=1 a i P p i=1 b i 1

. The Gamma funtion ~

q

(s), is losedrelated with

the lassialq Gamma funtion

q

(s) [25 ℄ denedby

q (s)= 8 > > > > > > > > > < > > > > > > > > > : (1 q) 1 s 1 Y k=0 (1 q k+1 ) 1 Y k=0 (1 q s+k )

; 0<q <1

q

(s 1)(s 2)

2

q

1(s); q>1

(8)

~

q (s)=q

(s 1)(s 2)

4

q

(s): (2.36)

Notie that ~

q

(s+1)=[s℄

q ~

q

(s), whereas

q

(s+1)= q

s

1

q 1 q

(s) =q s 1

2

[n℄

q q (s).

Theaboverepresentations(2.27)-(2.30) alsofollowfromtheEq. (2.24)substituting

(2.25) anddoing some operationssimilarto those in[37 ℄.

2.5 The orthogonality property and three-term reurrene relation.

2.5.1 The orthogonality relation

In[22 , 34, 37 ℄ hasbeenshownthat the polynomialsolutions P

n (x(s))

q P

n (s)

q of

thedierene equation (2.2) satisfyanorthogonalitypropertyof theform

b 1

X

s

i =a

P

n (x(s

i ))

q P

m (x(s

i ))

q (s

i

)4x(s

i 1

2 )=Æ

nm d

2

n ; s

i+1 =s

i

+1; (2.37)

where (x) is a solution of the Pearson-type equation (2.11) or (2.23), and it is a

non-negative funtion(weight funtion),i.e.,

(s

i

)4x(s

i 1

2

)>0 (as

i

b 1);

supportedona ountable subsetofthereal line[a;b℄(a;ban be1),providingthat

the ondition

(s)(s)x k

(s 1

2 )

s=a;b

=0; k =0;1;2;::: ;

(2.38)

holds[37 ℄. Infat,substitutingin(2.23)theexpressions(2.25),weobtainfortheweight

funtion(formally) theexpression

(s)=

q (s s

1 )

q (s s

2 )

q (s s

1 +1)

q (s s

2 +1)

;

i.e., (s)isa ratiooftheGamma funtions ~

q (s s

i

+1) and ~

q (s+s

i

+1) (i=1;2),

soit isa funtionwhih hasonlysimplepoles,and then we an always nda ontour

C inorder toapplythe Cauhyformulain(2.20).

In(2.37) d 2

n

denotes thesquare of thenorm of theorrespondingorthogonal

poly-nomials(with disreteorthogonalityrelations). In orderto alulateit we an usethe

expression dedued in [34 , Chapter 3, Setion 3.7.2, pag. 104℄ (see also [1 , Theorem

5.4℄)

d 2

n

=( 1) n

A

nn B

2

n b n 1

X

s=a

n

(s)4x

n (s

1

2

): (2.39)

If there existsaontour suh that, insteadof (2.38),the ondition

Z

4[(z)(z)x k

(z 1

2

)℄dz =0; 8k=0;1;2;::: ;: (2.40)

holds, then, the polynomials satisfy a ontinuous orthogonality relation of the form

[14 , 17 ,34 , 37 ℄

Z

P

n (z)P

m

(z)(z)4x(z 1

2

(9)

relation beomes an orthogonality relationon the real line. Usingthe above method,

AtakishiyevandSuslovprovedinaveryeasywaytheorthogonalityoftheAskey-Wilson

polynomials [17℄. In this work we will not onsidersuh a kind of orthogonality. For

more detailssee [11 , 13 ,14 ,17 , 26 ,37℄, among others.

2.5.2 The three-term reurrene relation.

Asimple onsequeneof (2.37) is thefatthatthe q-orthogonalpolynomialssatisfy

a three-term reurrenerelations (TTRR) oftheform

x(s)P

n (s)

q =

n P

n+1 (s)

q +

n P

n (s)

q +

n P

n 1 (s)

q

; (2.41)

withthe initialonditionsP

1 (s)

q

=0and P

0 (s)

q =1.

In order to alulate the oeÆients

n ,

n

, and

n

, we will substitute the

poly-nomial P

n (s)

q = a

n x

n

(s)+b

n x

n 1

(s)+in (2.41) (a

n

is usually alled the leading

oeÆient of the polynomial P

n (s)

q

). Then, equating the oeÆients for the powers

x n

(s) and x n 1

(s), we ndexpressions for

n and

n

, respetively. To obtain

n , we

are onstrainedto usetheorthogonalitypropertyof thepolynomialsP

n (s)

q ,

n =

b 1

X

s

k =a

x(s

k )P

n (s

k )P

n 1 (s

k )(s

k

)4x(s

k 1

2 )

d 2

n 1

:

So we obtaintheexpressions

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

: (2.42)

Sometimes, the omputation of b

n

and then the

n

an be very umbersome,so ifwe

know

n and

n andP

n

(a)6=0forall n, we an use(2.41) whihgivesus

n =

x(a)P

n

(a)

n P

n+1

(a)

n P

n 1 (a)

P

n (a)

:

3 Some onsequenes of the Rodrigues-type formula.

3.1 The onnetion between the leading oeÆient a

n

and the

Ro-drigues oeÆient B

n .

Firstof all,we willobtainanexpressionfortheleadingoeÆient a

n

ofthe

polyno-mial P

n (s)

q

. Notiethat

4x n

(s)

4x(s) =[n℄

q x

n 1

(s+ 1

2

)+; so 4 (n)

[x n

(s)℄=[n℄

q !:

Then, we have inone hand, 4 (n)

P (n)

n

(s)=[n℄

q !a

n

,and on theother, byusing(2.15),

4 (n)

P

n (s)

q =B

n A

nn

,sotaking into aount expression(2.16) we nallyobtain(a

0 =

B

0 )

a

n =B

n n 1

Y

k=0 (

q 1

2

(n+k 1)

+q 1

2

(n+k 1)

2

!

~ 0

+[n+k 1℄

q ~ 00

2 )

(10)

Letusnowobtainthedierentiation formulasfortheq-polynomials. From formulas

(2.11) and(2.12), we nd

5 n+1 (s) 5x n+1 (s) = 5[ n

(s+1)(s+1)℄

5x n (s+ 1 2 ) = 4[(s) n (s)℄ 4x n (s 1 2 ) = n (s) n (s):

Then byusingtheRodrigues-type formula(2.17), we obtain

P n+1 (s) q = B n+1 (s) 5 (n+1) [ n (s)℄= B n+1 (s) 5 (n) 5 n+1 (s) 5x n+1 (s) = = B n+1 (s) 5 (n) [ n (s) n (s)℄= B n+1 (s) Æ Æx(s) n [ n (s n 2 ) n (s n 2 )℄: (3.2)

Inordertoobtainanexpressionfor h Æ Æx(s) i n [ n (s) n

(s)℄,weanusetheq-analogue

of theLeibnitzformulaon thenon-uniformlatties

Æ Æx(s) n [f(s)g(s)℄= n X k =0 [n℄ q ! [k℄ q ![n k℄ q ! ( Æ Æx(s+ n k 2 ) k [f(s+ n k 2 )℄ )( Æ Æx(s k 2 ) n k [g(s k 2 )℄ ) ; (3.3)

whihan beproved byindution. Next, sine

Æ n (s n 2 + n 1 2 ) Æx(s+ n 1 2 ) = 0 n ; " Æ Æx(s+ n k 2 ) # k [ n (s n 2 + n k 2

)℄=0; 8k2;

then, Eq. (3.2) transforms

P n+1 (s) q = B n+1 (s) 0 n (s) Æ Æx(s) n [ n (s n 2

)℄+[n℄

q 0 n " Æ Æx(s 1 2 ) # n 1 [ n (s n 2 1 2 )℄ 1 A : (3.4)

Nowwe usetheRodrigues-typeformula(2.18)

5P n (s) q 5x(s) = 4P n (s 1) q 4x(s 1) = n B n 1 (s 1) 5 (n) 1 n

(s 1) =

= n B n (s)(s) " Æ Æx(s 1 2 ) # n 1 [ n (s n 2 1 2 )℄:

This leadsto theexpression,

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 5P n (s) q 5x(s) ;

and then, thefollowingdierentiation formulaholds

(11)

This proof generalizestheone presented in[3 ℄ to anylattie x(s)= 1 q + 2 q + 3 .

Notiethatforproving(3.5)wehave usedonlytheRodriges-typeformula(2.13).

For-mula (3.5) has been also obtained in [13℄ byusing the integral representation for the

q-polynomials aswellassome boundaryonditions.

Inordertoobtaintheseonddierentiationformula,weanmakeuseoftheidentity

4 5P n (s) q 5x(s) = 4P n (s) q 4x(s) 5P n (s) q 5x(s) : (3.6)

Then, byusingthediereneequation (2.2), theequation (3.5) gives

[(s)+(s)4x(s 1 2 )℄ 4P n (s) q 4x(s) = = n [n℄ q 0 n ( n (s) 4x(s 1 2 )[n℄ q 0 n )P n (s) q B n B n+1 P n+1 (s) q : (3.7)

3.3 The struture relations in the exponential lattie.

Let us obtain the struture relations for the polynomial solutions of (2.2) in the

lattie x(s)=

1 q

s

+

3

. Inorder todo thiswesubstitutethepowerexpansionof

n (s) n (s)= 0 n x n (s)+ n (0)= 0 n q n 2 x(s)+ n (0) 0 n 3 (q n 2 1);

in(3.5). Then,usingtheTTRR(2.41) we obtaintherst struture relation

(s) 5P n (s) q 5x(s) = ~ S n P n+1 (s) q + ~ T n P n (s) q + ~ R n P n 1 (s) q ; (3.8) where ~ S n = n [n℄ q q n 2 n B n 0 n B n+1 ; ~ T n = n [n℄ q q n 2 n + n (0) 0 n 3 (q n 2 1) ; ~ R n = n q n 2 n [n℄ q : (3.9)

Toobtaintheseondstruturerelationwetransform(3.8)withthehelpof(3.6). Then,

usingthefat that4x(s 1 2 )= q x(s) 3 q

,aswellasthediereneequation (2.2)

and theTTRR(2.41) one gets

[(s)+(s)4x(s 1 2 )℄ 4P n (s) q 4x(s) =S n P n+1 (s) q +T n P n (s) q +R n P n 1 (s) q ; (3.10) where S n = ~ S n n n q ; T n = ~ T n n n q + 3 n q ; R n = ~ R n n n q ; (3.11)

or, using(3.9),

(12)

Inthepresentsetionwewillprovethattheq polynomialsontheexponentiallattie

satisfya relationof theform

P

n (s)

q =L

n 4P

n+1 (s)

q

4x(s)

+M

n 4P

n (s)

q

4x(s) +N

n 4P

n 1 (s)

q

4x(s)

; (3.13)

where L

n ,M

n

and N

n

aresome onstants.

Toprove(3.13)weapplytheoperator 4

4x(s)

onbothsidesof(3.8),andthenusethe

seond orderdiereneequation (2.2)inwhihwewillhangetheoperator 4

4x(s 1

2 )

by

the equivalent one (only in the exponential lattie) q 1

2 4

4x(s)

. This is possible beause

fortheexponentiallattie thefollowingidentityholdsq 1

2

4x(s)=4x(s 1

2

). Thus,

q 1

2 4(s)

4x(s)

(s)

4P

n (s)

q

4x(s)

n P

n (s)

q =

=q 1

2 ~

S

n 4P

n+1 (s)

q

4x(s) +q

1

2 ~

T

n 4P

n (s)

q

4x(s) +q

1

2 ~

R

n 4P

n 1 (s)

q

4x(s) :

(3.14)

Next, we use the fat that (s) and (s) are polynomials of seond and rst degree,

respetively. Let

(s)=

00

2 x

2

(s)+ 0

(0)x(s)+(0); and (s)= 0

x(s)+(0): (3.15)

Sine, x(s +1) = qx(s)

3

(q 1), and using (3.15) we onlude that 4(s)

4x(s) is a

polynomialofrst degree inx(s). Then

q 1

2 4(s)

4x(s)

(s)

=Ax(s)+B;

where

A=

00

2

(1+q)q 1

2

0

; and B=q 1

2

0

(0)

00

2

3 q

1

2

(q 1) (0); (3.16)

and (3.14) beomes

Ax(s) 4P

n (s)

q

4x(s)

n P

n (s)

q =

=q 1

2 ~

S

n 4P

n+1 (s)

q

4x(s) +q

1

2 ~

T

n 4P

n (s)

q

4x(s) +q

1

2 ~

R

n 4P

n 1 (s)

q

4x(s)

B 4P

n (s)

q

4x(s) :

(3.17)

Now we use theTTRR (2.41) to eliminatethe term x(s) 4P

n (s)

q

4x(s)

. In fat, if we apply

theoperator 4

4x(s)

tobothsidesof(2.41) anduseagaintheidentityx(s+1)=qx(s)

3

(q 1),weget

qx(s) 4P

n (s)

q

4x(s) =

n 4P

n+1 (s)

q

4x(s)

+[

n +

3

(q 1)℄ 4P

n (s)

q

4x(s) +

n 4P

n 1 (s)

q

4x(s) P

n (s)

(13)

[A+q

n ℄P

n (s)

q =

h

n

A q

3

2 ~

S

n i

4P

n+1 (s)

q

4x(s) +

+ h

n A+

3

A(q 1)+qB q 3

2 ~

T

n i

4P

n (s)

q

4x(s) +

h

n

A q

3

2 ~

R

n i

4P

n 1 (s)

q

4x(s) ;

(3.18)

whihis oftheform (3.13) ifA+q

n 6=0.

To onlude thissetion notie that theexpression (3.13) an also be obtainedby

using, withsome modiations,themethod givenin [32 ℄.

4 The q-analogues of the lassial disrete polynomials

on the exponential lattie x(s)= q s

.

Inthissetionwewillonsidersomeq-analoguesoftheHahn,MeixnerandKravhuk

polynomials in the exponential lattie x(s) = q s

. These polynomials were onsidered

by other authors [19 , 39 ℄ using the Nikiforov et al. approah. In this setion we will

introdue them in an unify way from Eqs. (2.27)-(2.30) and their limits (see below

for more details), i.e., we willidentify these q-Hahn, q-Meixnerand q-Kravhuk with

ertain q-hypergeometri series whih have not been done in [19 , 39 ℄. The ase of a

q-analogueoftheCharlierpolynomialswillbeonsideredwithmoredetailsinthenext

setion.

4.1 The q-Hahn polynomials h ;

n

(s;N)

q .

Theq-Hahnpolynomials,from thepointof viewof[34 ℄,havebeenstudiedin[39 ℄. If

wehoosein(2.27) and(2.29) theparameters

x(s)=q s

!

1

=1;

3

=0; B

n =

( 1) n

q n

2

n

q [n℄

q !

;

s

1

=0; s

2

=N +; s

1

= 1; s

2

=N 1;

A= q N+

;

wereoverthe q-Hahnpolynomials [26 , 39 ℄

h ;

n

(s;N)

q =

(+1jq)

n

(1 Njq)

n

q n

2

(2++N+ n+1

2 )

[n℄

q !

3 F

2

n; s;n+++1

+1;1 N

;q;q 1

2

(s N ) !

=

=

(+1jq)

n

(N +++1jq)

n

q n

2 (N++

n(n+1)

2 )

[n℄

q !

3 F

2

n;s++1;n+++1

+1;N +++1

;q;q 1

2

(s+1 N) !

;

or,

h ;

n

(s;N)

q =

( 1) n

q n(+N)

(q +1

;q)

n (q

1 N

;q)

n

n

q (q;q)

n

3 '

2 q

n

;q s

;q

n+++1

q +1

;q 1 N

;q;q

s N +1 !

=

= (q

+1

;q)

n (q

N+++1

;q)

n

q n

2

(2+n+1)

n

q (q;q)

n 3

'

2 q

n

;q s++1

;q

n+++1

q +1

;q

N+++1

;q;q !

(14)

n

n =q

1

2

(++2)

[n℄

q

[n+++1℄

q ;

and (2.37) holdsintheinterval [0;N 1℄where

(s)=q

4

(+2N+2s 3)+

4

(+2s 1) ~

q

(+N s) ~

q

(+s+1)

[N s 1℄

q ![s℄

q !

; ; 1; nN 1;

and

d 2

n =

q 1

2

N(N 1)+ 1

2

(N 1)(2++N)

q N

1

4 (+1)

1

2

n(+ 2)

q ~

q

(+n+1) ~

q

(+n+1) ~

q

(++N +n+1)

[n℄

q

![N n 1℄

q !

~

q

(++n+1) ~

q

(++2n+2) :

4.2 The q-Meixner, q-Kravhuk polynomials.

From (2.27) or (2.29), and taking the limits q s

i

! 0 or q s

i

! 0, i = 1;2, it is

possibleto nddierentfamiliesofq-polynomials. In[37℄,9dierentpossibilitieshave

beenonsidered. Herewewillstudyonly2ofthem,i.e., theasesorrespondingtothe

q-Meixner, q-Kravhuk, respetively. The q-analogue of the Charlier polynomial will

be introduedin thenext setion.

Taking the limitq s

2

! 0; q s

2

! 0, and hoosing

A =A

1 q

s

2

, s

2 s

2

=Æ+1, the

expression(2.25) beomes

(s)=A

1 q

s

(q s s

1

1); (s)+(s)4x(s 1

2 )=A

1 q

Æ+1

q s

(q s s

1

1): (4.1)

Then, (2.27) and (2.29) leadusto

P

n (s)

q =

A

1

1

n

B

n q

n(n+1)

2 +nÆ

(s

1 s

1 jq)

n

3 F

1

n;s

1 s

1

+Æ+n;s

1 s

s

1 s

1

;q;q 1

2

(s s1 Æ 1) !

;

(4.2)

and

P

n (s)

q =B

n 0

A

1 q

(n 1)

4 +Æ+1

1 (q

1

2

q 1

2

) 1

A n

(q s1 s1

;q)

n 3 '

1 q

n

;q s

1 s

1 +Æ+n

;q s

1 s

q s1 s1

;q;q s s1 Æ

!

;

respetively. Inthisase, takinglimitson theexpression(2.26) we nd

n =

A

1

2

1 q

1

2 (s

1 +s

1 Æ 1)

[n℄

q [s

1 s

1

+Æ+n℄

q

: (4.3)

4.2.1 The q-Meixner polynomials m ;

n

(s;q).

LetA

1

=1,B

n =

n

,s

1

=0, s

1

= ,q Æ

=. Then Eqs.(4.2) and (4.2) dene a

q-analogueof theMeixner polynomials

m ;

n

(s;q)=q n(n+1)

2

(jq)

n3 F

1

n;+Æ+n; s

;q;q 1

2 (s Æ 1)

!

(15)

m ;

n

(s;q)= ( 1) n

q

(n+3)

4

(q 1

2

q 1

2

) A

n

(q

;q)

n3 '

1 q

n

;q +n

;q s

q

;q; 1

q s

!

:

In thisase(4.1) and (4.3) beome

(s)=q s

(q s

1); (s)+(s)4x(s 1

2 )=q

s+1

(q s+

1); (4.4)

and

n =[n℄

q q

(n 1)

2

q (n+1)

2 +

q

; (4.5)

respetively. Finally,thePearson-type equation leadsusto theweight funtion

(s)=

s

q (+s)

q ()

q (s+1)

; >0; 0<<1;

sothat, (2.37) holdsin theinterval[0;1)and

d 2

n =q

1

2 n(6 n)

n+1

q

(jq)

n q

(+Æ+2n)

q

(n+1)( q 2n++1

;q)

1

n

q

(+Æ+n)( q n+1

;q)

1

:

TheseMeixnerq polynomialsoinidewiththosestudiedin[19 ℄. Notiealsothatthese

q analogues of the Meixner polynomials m ;

n

(s;q) are theq-little Jaobip

n

(x;a;bjq)

polynomials[25 , 26 ℄, i.e., m ;

n

(s;q)=p

n (q

s

;;q 1

jq).

4.2.2 The q-Kravhuk polynomials k (p)

n

(s;q).

Analogously,from(4.2) and (4.2), butnowhoosing

A

1

= 1; B

n

=( 1) n

(1 p) n

[n℄

q !

; s

1

=0; s

1

=N; q Æ

= p

1 p q

N

;

wenda q-analogueof theKravhukpolynomials

k (p)

n

(s;q)=( 1) n

p n

q n(n+1)

2 +Nn

( Njq)

n

[n℄

q !

3 F

1

n; N+Æ+n; s

N

;q;q 1

2 (s Æ 1)

!

;

k (p)

n

(s;q)= ( 1) n

pq (n+3)

4 +N

n

(q N

;q)

n

(q;q)

n 3

'

1 q

n

; p

p 1 q

n

;q s

q N

;q; p 1

p q

N+s !

:

Notie that

( 1) n

pq (n+3)

4 +N

n

(q N

;q)

n

(q;q)

n =

pq 3(n+1)

4

n

q

(N +1)

[n℄

q !

q

(N n 1)

:

In thisasewehave

(s)=q s

(1 q s

); (s)+(s)4x(s 1

2 )=

p

1 p q

s+1+N

(1 q s N

);

(4.6)

n

= [n℄

q p

1 p q

(n+1)

2

+q (n 1)

2

q

(16)

(s)=

p

1 p

s

q 1

2 s(s+1)

[N℄

q !

q

(N +1 s)

q (s+1)

; 0<p<1;

and

d 2

n =q

1

2

n(n++2)

q p

n

(1 p) n

[n+ 1℄

q

![4n+2 2℄

q !!(

p

1 p q

2n+1

;q)

(N n)

[n℄

q

![2n+ 1℄

q

![2n+2 2℄

q !!

q

(N n+1)

; q

= p

1 p :

These polynomials wherestudiedindetail in[19 ℄.

5 q-Charlier polynomials.

Intheprevioussetionwehaveobtainedsomeq analoguesoftheHahn,Meixnerand

Kravhuklassialpolynomialsonthegeneralexponentiallattiewithspeialemphasis

in thelattie x(s)=q s

. Here we willonsiderthe fourth family,i.e., a q analogue of

theCharlierpolynomials.

Firstly, we take the limits q s

2

! 0; q s

1

! 0; q s

2

! 0, and hoose the others

parameters as

A=A

1 q

s

2

,s

2 s

1

=Æ+1+ i

logq

. Then,(2.25) beomes

(s)=A

1 q

s

(q s s

1

1); (s)+(s)4x(s 1

2 )=A

1 q

Æ

q s+1

; (5.1)

and (2.27) and (2.29) transforms into

P

n (s)

q =

A

1

q

1 !

n

B

n q

n(n+3)

4 +nÆ

2 F

0

n;s

1 s

;q; q 1

2 (s s

1

n 1) Æ !

; (5.2)

P

n (s)

q =

A

1

q

1 !

n

B

n q

n(n+3)

4 +nÆ

2 '

0 q

n

;q s1 s

;q; q s s

1 Æ

!

; (5.3)

Finally,from (2.26) one gets

n =

A

1

q

2

1 q

s1 n 1

2

[n℄

q

: (5.4)

5.1 The q-Charlier polynomials in the lattie x(s)=q s

.

Letusnowintroduetheq-Charlierpolynomialsinthelattiex(s)=q s

. Inorderto

do thatwehoose

x(s)=q s

; A

1

=1; B

n =

n

; s

1

=0; q Æ

=(q 1):

Then, formulas (5.2) and (5.3) give us

()

n

(s;q)=q n(n+5)

4

2 F

0

n; s

;q; q

1

2

(s n 1)

(q 1) !

;

()

n

(s;q)= q n(n+5)

4

2 '

0 q

n

;q s

;q; q

s

(q 1) !

;

(17)

(s)=q (q 1); (s)+(s)4x(s 1

2

)=(q 1)q ; (5.5)

and

n =

q n 1

2

q [n℄

q

: (5.6)

Finally,fortheweight funtion(s), thePearson-type diereneequation (2.23) gives

(s)=

s

e

q

[(1 q)℄

q (s+1)

; 0<(1 q)<1; (5.7)

where e

q

[z℄denotestheq- exponentialfuntion[25 ℄ denedby

e

q [z℄=

1

X

k=0 z

k

(q;q)

k =

1

(z;q)

1

; and (z;q)

1 =

1

Y

k=0 (1 zq

k

): (5.8)

These polynomials werepartiallystudied in[1 ℄.

Main harateristis of the q Charlier polynomials in the lattie x(s)=q s

.

Usingthe formulas obtained inthe above Setions we an ndthe mainproperties

of theq-Charlierpolynomials. Theyaregiven inTable 1.

Table 1: Maindatafortheq-Charlierpolynomials inthelattie x(s)=q s

.

Pn(s)q

()

n

(s;q); x(s)=q s

(a;b) (0;1)

(s)

s

eq[(1 q)℄

q (s+1)

; >0; 0<(1 q)<1

(s) q

s

(q s

1)

(s)

1

q

q(q 1)+1 1

q q

s

n(s)

q n

q xn(s)+

q n+

3

2

q +1

q n

2

q

n [n℄qq

(n 1)

2

1

q

Bn

n

d 2

n

(f1 qg;q)

n+1

q[n℄q!

q n

4 (n 9)

n

=

eq[(1 q)q n+1

e

q

[(1 q)℄

q[n℄q!

q n

4 (n 9)

n

n(s)

s+n

n

q q

n

2 (n+2s+2)

eq[(1 q)℄ q(s+1)

a

n

( 1) n

n

q

n q

3n

4 (n 1)

Expliit Formula.

()

n

(s;q)= ( 1)

n

[n℄

q !

n

n

X

m=0 ( 1)

m

q m

2 (n+1)

n

4 (n 3)

m

q (s+1)

[m℄

q

![n m℄

q !

q

(18)

Table1: Main datafortheq-Charlierpolynomialsinthelattie x(s)=q (ont.)

Pn(s)q

()

n

(s;q); x(s)=q s

n

q q

3

2 n

n 1+qq 2n+

3

2 n

+q 3(n+1)

2

[n℄q( 1 (1 q)q n

) o

n q

n+1

q[n℄qf1 (1 q)q n

g

~

S

n

q n+1

2

(1 q n

)

~

Tn [n℄qq n+1

2

f 1 q n

(1 q)g q n+2

( 1 q n

)

~

R

n

q n+

3

2

[n℄

q

(1 (1 q)q n

)

Sn 0

Tn (1 q)[n℄qq

n+1

2

Rn q

3

2

[n℄q( 1 (1 q)q n

)

L

n

q n

q

[n+1℄

q

Mn

q n+1

2

(q n+1

1)q

[n+1℄q

N

n

0

Speial Values

()

n

(0;q) =q n

4 (n+5)

:

Dierentiation formulas.

(q s

1)5 ()

n

(s;q)=(q 1)q n 1

2

()

n+1 (s;q)

h

1+(q 1)q n+1

q s

i

()

n (s;q):

(q 1)4 ()

n

(s;q)=q n 1

2

()

n+1 (s;q)

h

1+(q 1)q n+1

+q s n

i

()

n (s;q):

Notie that if we take the limit q ! 1, we willnot reover the main data for the

lassialCharlierpolynomials. Forthisreasonwewillintrodueadierentq analogue

of theCharlierpolynomials,butinthe lattie x(s)= q

s

1

q 1 .

5.2 The q Charlier polynomials in the lattie x(s)= q

s

1

q 1 .

To avoid theproblemommentedbelow,we hoosenow

x(s)= q

s

1

q 1 ; A

1 =

1

q 1 ;

1

=

3 =

1

q 1 B

n =

n

; s

1

=0; q Æ

=(q 1):

(19)

Table 2: Maindata fortheq-Charlierpolynomialsinthelattie x(s)= q 1

q 1 .

Pn(s)q

()

n

(s)q; x(s)= q

s

1

q 1

(a;b) [0;1)

(s)

s

eq[(1 q)℄

q (s+1)

; >0; 0<(1 q)<1

(s) q

s

x(s)

(s) q

3

2

q 1

2

x(s)

n(s) q

n+ 1

2

xn(s)+q n

2 +

3

2

+q 3

4 n

[ n

2 ℄q

n [n℄qq

(n 2)

2

Bn

n

d 2

n

(f1 qg;q)

n+1

[n℄q!

q n

4 (n 9)+

1

2

n

=

eq[(1 q)q n+1

eq[(1 q)℄

[n℄q!

q n

4 (n 9)+

1

2

n

n (s)

s+n

q n

2

(n+2s+1)

e

q

[(1 q)℄

q (s+1)

a

n

( 1) n

n

q 3n

4 (n 1)+

n

2

()

n (s)

q =q

n(n+5)

4

2 F

0

n; s

;q; q

1

2

(s n 1)

(q 1) !

;

and

()

n (s)

q = q

n(n+5)

4

2 '

0 q

n

;q s

;q; q

s

(q 1) !

;

respetively. Thefuntions (s) and (s) and

n

are denednowbyrelations

(s)=q s

q s

1

q 1

; (s)+(s)4x(s 1

2 )=q

s+1

;

n =q

n

2 +1

[n℄

q

: (5.9)

Finally,fortheweight funtion(s), thePearson-type diereneequation (2.11) gives

(s)=

s

e

q

[(1 q)℄

q (s+1)

; 0<(1 q)<1: (5.10)

where e

q

[z℄wasgiven in(5.8).

Notiethatwiththepresenthoieofparametersthehypergeometrirepresentation

remainsthesame buttheoeÆientsoftheseondorderdiereneequation(s),(s)

and

n

tend, when q ! 1, to the orresponding oeÆients of the lassial Charlier

polynomials,and thethelattie funtionx(s)= q

s

1

q 1

beomesthelinear one.

Mainharateristis ofthe q Charlierpolynomialsinthe lattiex(s)= q

s

1

q 1 .

Again,usingtheformulasobtainedintheabove Setionsweobtainthepropertiesof

(20)

Table 2: Maindatafortheq-Charlier polynomialsin thelattie x(s)= q 1

q 1

(ont).

Pn(s)q

()

n

(s)q; x(s)= q

s

1

q 1

n

q 3

2 n

1

2

n q

2n+1

+[n℄qf 1 (1 q)q n

gq n 1

2

n q

n+ 1

2

[n℄qf1 (1 q)q n

g

~

S

n

q n+1

2

(1 q n

)

~

T

n [n℄

q q

n+1

2

f1 (1 q)q n

g q n+2

(1 q n

)

~

Rn q

n+ 3

2

[n℄qf1 (1 q)q n

g

S

n

0

Tn (1 q)[n℄qq

n+1

2

R

n

q 3

2

[n℄

q

f 1 (1 q)q n

g

Ln

q n

1

2

[n+1℄

q

Mn

q n

2

(q n+1

1)

[n+1℄q

Nn 0

Expliit Formula.

()

n

(s;q)= ( 1)

n

[n℄

q !

n

n

X

m=0 ( 1)

m

q m

2 (n+1)

n

4 (n 3)

m

q (s+1)

[m℄

q

![n m℄

q !

q

(s+1 n+m) :

SpeialValues

()

n (0)

q =q

n

4 (n+5)

:

Dierentiation formulas.

q s

1

q 1 5

()

n

(s;q)=q n 1

2

()

n+1 (s;q)

q n+1

q s

1

q 1

()

n (s;q):

4

()

n

(s;q)=q n 1

2

()

n+1

(s;q)+ "

q s n

1

q 1 q

n+1 #

()

n

(s;q):

Aswealreadypointedout,allharateristisofthese q Charlierpolynomialstend

to the orresponding harateristis of the lassial ones when q tends to 1, whih

doesn'thappensinthepreviousase(lattiex(s)=q s

,seeTable1). So,these

polyno-mials are more natural q analogues of the lassial ones, and the exponential lattie

x(s) = q

s

1

q 1

is a more natural lattie than the previous one x(s) = q s

. Moreover,

(21)

theapproahbasedonthehypergeometriseries[26℄). Butifweonsiderthemas

fun-tionsofx(s)(i.e.,aspolynomialsinx(s)),weobtaintwoompletedierentfamiliesand

one of them (theone on x(s)= q

s

1

q 1

) seemto bea more naturalq analoguethanthe

other one. For thisreason it is onvenient to omplete the study of the q analogues

of lassial disrete polynomials in the lattie x(s) = q

s

1

q 1

. This will be done in a

forthoming paper(see [7℄).

Aknowledgements

Thiswork wasompletedwhileone oftheauthor(RAN) wasvisitingtheUniversidade

de Coimbra. He is very gratefulto theDepartment of Mathematis of Universidadde

Coimbra for the kind hospitality and the Centro de Matematia da Universidadede

Coimbra for nanial support. The authors are very grateful to Professors N.

Atak-ishiyevandF.Marellanwhohavearefullyreadthismanusriptandhelpedtoimprove

it and also to Professor S. Lewanowitz who help us to orret some missprints. The

researh of the authors was partially supported by Direion General de Ense~nanza

Superior(DGES) PB96-0120-C03-01 and theEuropean projetINTAS-93-219-ext.

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Figure

Table 1: Main data for the q-Charlier polynomials in the latti
e x(s) = q (
ont.) Pn(s)q 
 () n (s; q) ; x(s) = q s  n  q q 32 n n 1 + qq 2n+ 32 n  + q 3(n+1)2 [n℄q ( 1 (1 q)q n ) o 
n q n+1 q[n℄qf1 (1 q)q n g ~ S n q n+12 (1 q n ) ~ Tn [n℄qq n+

Referencias

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