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
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
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
(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
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
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 )
!
;
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
~
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
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 )
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
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),
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)
[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 !
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)
!
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
(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) !
;
(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
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):
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
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,
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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