R.
Alvarez-Nodarse
,M. K.Atakishiyevay, and N.M. Atakishiyevz
Departamento deAnalisisMatematio, Universidadde Sevilla,
Apdo. 1160, E-41080Sevilla,Spain and
InstitutoCarlos Ide Fsia Teoria yComputaional,
Universidadde Granada, E-18071 Granada, Spain
E-mail: ranus.es
yFaultadde Cienias,UAEM, ApartadoPostal396-3,
CP62250, Cuernavaa, Morelos, Mexio
E-mail: mesumaservm.f.uaem.mx
zInstituto deMatematias, UNAM, ApartadoPostal273-3,
C.P.62210Cuernavaa, Morelos, Mexio
E-mail: natigmatuer.unam.mx
Abstrat
Westudyapolynomialsequeneofq-extensionsofthelassialHermitepolynomials
H
n
(x), whih satises ontinuous orthogonality on the whole real line R with respet
to the positiveweight funtion. This sequene anbe expressed either in termsof the
q-LaguerrepolynomialsL ()
n
(x;q),=1=2,orthroughthedisreteq-Hermite
polyno-mials ~
h
n
(x;q)oftypeII.
1 Introdution
Thereis awell-knownfamilyof theontinuousq-Hermite polynomialsofRogers [1, 2℄
H
n
(osjq):= n
X
k=0
n
k
q e
i(n 2k)
=e in
2
0 q
n
;0
|
q;q
n
e 2i
!
; (1.1)
whihareq-extensionsofthelassialHermitepolynomialsH
n
(x)forq2(0;1). Throughout
thispaperwewillemploythestandardnotationsoftheq-speialfuntionstheory,see[3 ℄-[5 ℄.
In partiular,
n
k
q :=
(q;q)
n
(q;q)
k (q;q)
n k =
n
n k
q
(1.2)
stands for the q-binomial oeÆient and (a;q)
0
= 1 and (a;q)
n =
Q
n 1
j=0
(1 aq j
);n =
1;2;3;:::,is theq-shiftedfatorial. Besides,expliitformsofq-polynomialsfrom the
Askey-sheme [4 ℄ areoftenexpressed intermsof theterminatingbasihypergeometri polynomial
r
s
q n
;a
2 ;:::;a
r
b
1 ;b
2 ;:::;b
s
q;z
:= n
X
k=0 (q
n
;q)
k (a
2 ;q)
k (a
r ;q)
k
(b
1 ;q)
k (b
2 ;q)
k (b
s ;q)
k z
k
(q;q)
k h
( 1) k
q
k(k 1)=2 i
s r+1
(1.3)
ofdegree ninthevariable z. So theontinuousq-Hermite polynomialsin(1) orrespond to
thease whenr =2 ands=0.
Theontinuousq-HermitepolynomialsH
n
(xjq)satisfythethree-term reurrenerelation
2xH
n
(xjq)=H
n+1
(xjq)+(1 q n
)H
n 1
(xjq): (1.4)
Theyare orthogonal on the niteinterval x2[ 1;1℄with respetto theontinuousweight
funtion[2 ℄
w(x)= 1
p
1 x 2
1
Y
k=0 h
1+2(1 2x 2
)q k
+q 2k
i
: (1.5)
Inthelimitas q!1,they oinide withtheordinaryHermitepolynomialsH
n
(x), i.e.,
lim
q!1
1 q
2
n=2
H
n r
1 q
2 x
q
!
=H
n
(x): (1.6)
There is another family, alled the disrete q-Hermite polynomials of type II , whih is
a q-extension of the sequene of the Hermite polynomials H
n
(x) [6 ℄. Their expliitform is
given(see [4, p. 119℄) by
~
h
n
(x;q):=i n
q
n(n 1)=2
2
0 q
n
;ix
|
q; q
n !
=x n
2
1 q
n
;q 1 n
0
q
2
; q
2
x 2
!
: (1.7)
The disrete q-Hermite polynomials of type II, ~
h
n
(x;q), satisfy the three-term reurrene
relation
x ~
h
n
(x;q)= ~
h
n+1
(x;q)+q 1 2n
(1 q n
) ~
h
n 1
(x;q); (1.8)
but,ontrary to thepolynomialsH
n
(xjq) of Rogers (1.1), thesequenef ~
h
n
(x;q)g is
orthog-onalontheinniteintervalx2R withrespettothedisreteweight funtion,supportedon
thepoints x=q k
;>0;k 2Z. In thelimit asq !1, the ~
h
n
(x;q) redue aswellto the
HermitepolynomialsH
n (x):
lim
q!1 (1 q
2
) n=2
~
h
n (x
p
1 q 2
;q)=2 n
H
n
(x): (1.9)
Now we are in a position to formulate the aim of this paper. We wish to nd suh
q-extensions oftheHermite polynomials H
n
(x), whih satisfythefollowingrequirements:
1. They are polynomials in the variable x, whih obey a three-term reurrene relation; 2.
They are orthogonal on the whole real line R with respet to a ontinuous positive weight
funtion;3. In thelimitasq !1they oinide withtheHermitepolynomialsH
n (x).
Suha familyisofgreat interestfrom thepointofview ofpossibleappliationsin
math-ematialphysis. We remind thereaderthat thetwo mostfundamentalproblems in
nonrel-ativisti quantum mehanis, the harmoniosillator and the Coulomb system, are dened
on R 3
[7 ℄. Thus our goal is equivalent to having suh a q-deformed version of the linear
harmoniosillator, whih isstilldenedonthe wholereallineR and enjoystheontinuous
orthogonalitypropertyon R withrespetto a positiveweight funtion. 2
We willnotpursue
thisviewpointhere. Insteadwenowfousonthemathematialpropertiesoftheq-extensions
ofthe Hermitepolynomials H
n
(x)under disussion.
2
Itshouldbenotedthatthereare,infat,severalpubliations(see[8 ℄{[16℄andreferenestherein)devoted
to the study of expliit realizations, whih represent q-extensions of the Hermite funtions (or the wave
funtionsof the linearharmoni osillator) H
n (x)e
x 2
=2
. Butnone of these realizations satises all of the
aforementioned requirements : the ontinuous weight funtions in [8, 10 , 13 ℄ are supported on the nite
intervals;theontinuousweightfuntionsin[9 ,15 ℄are notpositive;theq-extensionsin[9℄, [12 ℄{[15℄arenot
2 Denition of the sequene fH
n
(x;q)g
It is known that the Hermite polynomials H
n
(x) an be expressed through the Laguerre
polynomialsL ()
n
(x)as
H
2n
(x)=( 1) n
2 2n
n!L ( 1=2)
n (x
2
);
H
2n+1
(x)=( 1) n
2 2n+1
n!xL (1=2)
n (x
2
) ;
(2.1)
where
L ()
n
(z) :=
(+1)
n
n! 1
F
1
n
+1
z
=
(+1)
n
n! n
X
k=0 ( n)
k
(+1)
k z
k
k!
; (2.2)
and (a)
n
= (a+n)= (a); n=0;1;2;:::,istheshifted fatorial.
It is also known that a q-extension of the Laguerre polynomials 3
L ()
n
(x;q), dened
[17 ℄-[19℄as
L ()
n
(x;q)= (q
+1
;q)
n
(q;q)
n 1
1 q
n
q +1
q; q
n++1
x !
= 1
(q;q)
n 2
1 q
n
; x
0
q;q
n++1 !
;
(2.3)
satisestwokindsoforthogonalityrelations,anabsolutelyontinuousoneandadisreteone.
The formerorthogonalityrelation 4
is given by
1
Z
0 x
E
q (x)
L ()
m
(x;q)L ()
n
(x;q)dx=d 1
n ()Æ
mn
; > 1; (2.4)
whereE
q
(x) istheJakson q-exponentialfuntion,
E
q (z):=
1
X
n=0 q
n(n 1)=2
(q;q)
n z
n
=( z;q)
1
; (2.5)
and thenormalization onstant d
n
()is equalto
d
n
()=q n
(q;q)
n
(q +1
;q)
n
(q;q)
1
(q
;q)
1
sin(+1)
: (2.6)
It remainsonlyto remindthereader, that inthelimitasq!1 we have
lim
q!1 L
()
n
((1 q)x;q)=L ()
n
(x): (2.7)
3
Therearealsotheontinuousq-LaguerrepolynomialsP ()
n
(xjq)withtheorthogonalityontheniteinterval
x2[ 1;1℄andtheWall(orthelittleq-Laguerre)polynomialsp
n
(x;ajq)withthedisreteorthogonalityon
thepointsx=q k
;k=0;1;2;:::(see[4 ℄,pp. 105and107,respetively).
4
Itisworthnotingthatthefatoftheintegrabilityoftheweightfuntionx
=E
q
(x)in(2.4)diretlyfollows
fromtheintegral
1
Z
0 t
x 1
dt
Eq((1 q)t) =
(x) (1 x)
q(1 x)
We an now dene, in omplete analogy with the relationship (2.1), a family of
q-extensions oftheHermite polynomials oftheform
H
2n
(x;q):=( 1) n
(q;q)
n L
( 1=2)
n (x
2
;q);
H
2n+1
(x;q):=( 1) n
(q;q)
n xL
(1=2)
n (x
2
;q):
(2.8)
Withtheaidof thelimitrelation
lim
q!1 (q
a
;q)
n
(1 q) n
=(a)
n
(2.9)
itis notdiÆultto verifythat from(2.8) and (2.3) one obtains
lim
q!1 (1 q)
n=2
H
n (
p
1 qx;q)=2 n
H
n
(x); (2.10)
i.e.,thepolynomialsH
n
(x;q) sodenedare indeedq-extensionsof theHermitepolynomials
H
n
(x). The next step is to establish a ontinuous orthogonality relation and a three-term
reurrenerelationfortheq-polynomials H
n (x;q).
3 Orthogonality relation
We beginthissetion withthefollowingtheorem:
Theorem 1 Thesequeneoftheq-polynomialsfH
n
(x;q)g,whiharedenedbytherelations
(2.8), satises the orthogonality relation
1
Z
1 H
m
(x;q)H
n (x;q)
dx
E
q (x
2
) =q
n
2
(q 1=2
;q 1=2
)
n (q
1=2
;q)
1=2 Æ
mn
(3.1)
on the whole real line R with respet to the ontinuous positive weight funtion w(x) =
1=E
q (x
2
).
Proof. Sine the weight funtionin (3.1) is an even funtionof the independent variable x
andH
n
( x;q)=( 1) n
H
n
(x;q)bythedenition(2.8),the q-polynomialsofan even degree
H
2m
(x;q) and of an odd degree H
2n+1
(x;q); m;n = 0;1;2;:::, are evidentlyorthogonal to
eah other. Consequently, it suÆes to prove only those ases in (3.1), when degrees of
polynomialsm and nareeithersimultaneouslyevenorodd. Letusonsiderrst theformer
ase. From (2.8) and (2.4) itfollows that
1
Z
1 H
2m
(x;q)H
2n (x;q)
dx
E
q (x
2
)
=2( 1) m+n
(q;q)
m (q;q)
n 1
Z
0 L
( 1=2)
m (x
2
;q)L ( 1=2)
n (x
2
;q) dx
E
q (x
2
)
=2( 1) m+n
(q;q)
m (q;q)
n 1
Z
0 L
( 1=2)
m (x
2
;q)L ( 1=2)
n (x
2
;q) dx
E
q (x
2
)
=( 1) m+n
(q;q)
m (q;q)
n 1
Z
0 L
( 1=2)
m
(y;q)L ( 1=2)
n
(y;q) y
1=2
dy
E
q (y)
=
(q;q) 2
n
d ( 1=2) Æ
mn =q
n
(q 1=2
;q 1=2
)
2n (q
1=2
;q)
1=2 Æ
mn ;
where we have used at the last step the equality (q 1=2
;q)
n (q;q)
n = (q
1=2
;q 1=2
)
2n
and the
standardrelation(for a=q 1=2
and =1=2)
(a;q)
1
(aq
;q)
1
=(a;q)
; (3.3)
whih isvalidforan arbitraryomplex.
In thesame waywe nd
1
Z
1 H
2m+1
(x;q)H
2n+1 (x;q)
dx
E
q (x
2
)
=2( 1) m+n
(q;q)
m (q;q)
n 1
Z
0 L
(1=2)
m (x
2
;q)L (1=2)
n (x
2
;q) x
2
dx
E
q (x
2
)
=2( 1) m+n
(q;q)
m (q;q)
n 1
Z
0 L
(1=2)
m
(y;q)L (1=2)
n
(y;q) y
1=2
dy
E
q (y)
= (q;q)
2
n
d
n (1=2)
Æ
mn =q
(n+1=2)
(q 1=2
;q 1=2
)
2n+1 (q
1=2
;q)
1=2 Æ
mn ;
(3.4)
wherewehave used therelation
(1 q 1=2
)(q 3=2
;q)
n
=(1 q n+1=2
)(q 1=2
;q)
n
; (3.5)
whih is a straightforward onsequene of the general denition of the q-shifted fatorial
(a;q)
n
. Putting(3.2) and (3.4) together resultsintheorthogonalityrelation(3.1).
The positivity of Jakson's q-exponential funtion E
q (x
2
) for x 2 R and q 2 (0;1) is
obviousfromits denition(2.5): foritisrepresentedasaninnite sumofpositiveterms(or
an inniteprodut ofpositive fators). Thisompletes theproof.
To onludethissetion,we notetheobvious fatthat inthelimitasq !1 the
orthog-onality relation(3.1) reduesto thewell-knownorthogonalityproperty
1
Z
1 H
m (x)H
n (x)e
x 2
dx= p
2 n
n!Æ
mn
(3.6)
oftheHermitepolynomialsH
n
(x)withrespettothenormaldistributione x
2
. Thisfollows
immediatelyfrom thelimitrelations (2.9) and (2.10),upon usingthefatthat
lim
q!1 E
q
((1 q)z)=e z
: (3.7)
4 Reurrene relation
The easiest wayto identifythe sequene (2.8) ( that is, to nd its appropriate nihe inthe
Askey-sheme of q-polynomialfamilies[4 ℄ ),is to derive a three-term reurrenerelation for
it. Sinean arbitrarypolynomialp
n
(x)satisesa reurrenerelationof theform(see [21 ,p.
19℄)
(a
n x+b
n )p
n
(x)=p
n+1 (x)+
n p
n 1
(x); n0; (4.1)
one an try to evaluate the right-hand side of (4.1) for p
n
(x) = H
n
(x;q) and to nd then
Before starting this derivation we note that in what follows it proves onvenient to use
thefollowingform
L ()
n
(x;q)= (q +1 ;q) n (q;q) n n X k=0 q k(k+) (q +1 ;q) k n k q ( x) k (4.2)
of theq-Laguerre polynomials L ()
n
(x;q), whih omes from therst linein denition(2.3),
uponusingtherelation
(q n ;q) k (q;q) k
=( 1) k
q
k(k 1)=2 nk n k q : (4.3)
Let usrst onsiderthease when nin (4.1) is even. Then from (2.8) and (2.3) we nd
that
H
2n+1
(x;q)+
2n (q)H
2n 1 (x;q)
=( 1) n (q 3=2 ;q) n x n X k=0 q k(k+1=2) (q 3=2 ;q) k n k q ( x 2 ) k + 2n
(q)( 1) n 1 (q 3=2 ;q) n 1 x n 1 X k=0 q k(k+1=2) (q 3=2 ;q) k n 1 k q ( x 2 ) k : (4.4)
Thenextstepistoemploytherelation(3.5)inordertorewritethequotient(q 3=2 ;q) n =(q 3=2 ;q) k
fromthe rsttermin (4.4) as
(q 3=2 ;q) n (q 3=2 ;q) k = 1 q n+1=2 1 q k+1=2 (q 1=2 ;q) n (q 1=2 ;q) k : (4.5)
Intheseondtermin(4.4)oneanusetheevidentrelation(q 3=2 ;q) n 1 =(q 1=2 ;q) n =(1 q 1=2 )
and the same formula (3.5) for the fator (q 3=2
;q)
k
. Also, we reall the propertyof the
q-binomialoeÆient n 1 k q = 1 q n k 1 q n n k q : (4.6)
Puttingthisall together, we obtain
H
2n+1
(x;q)+
2n (q)H
2n 1 (x;q)
=( 1) n (q 1=2 ;q) n x n X k=0 q k(k+1=2) (q 1=2 ;q) k n k q ( x 2 ) k 1 q k+1=2 1 q n+1=2 2n (q) 1 q n k 1 q n : (4.7)
Theright-hand sideof(4.7) shouldmath with
H
2n
(x;q)=( 1) n (q 1=2 ;q) n n X k=0 q k(k 1=2) (q 1=2 ;q) k n k q ( x 2 ) k ; (4.8)
multipliedbya
2n
(q)x+b
2n
(q). ThismeansthattheoeÆient
2n
(q)an befoundfromthe
equation 1 q n+1=2 2n (q) 1 q n k 1 q n =d n (q)q k (1 q k+1=2
); (4.9)
whered
n
(q) is some k-independent fator. It is notdiÆult to verify that theonlysolution
ofthe equation(4.9) is
2n
(q)=1 q n
and d
n (q)=q
n
. Thus
H (x;q)+(1 q n
)H (x;q)=q n
Similarly,inthease of anodd nfrom (4.8) we have
H
2n+2
(x;q)+
2n+1 (q)H
2n (x;q)
=( 1) n+1 (q 1=2 ;q) n+1 n+1 X k=0 q k(k 1=2) (q 1=2 ;q) k
n+1
k q ( x 2 ) k + 2n+1 (q)( 1) n (q 1=2 ;q) n n X k=0 q k(k 1=2) (q 1=2 ;q) k n k q ( x 2 ) k : (4.11)
InthisaseitiseveneasiertondtheoeÆient
2n+1
(q). Indeed,onewillobtainfrom(4.11)
anexpression[a
2n+1
(q)x+b
2n+1 (q)℄H
2n+1
(x;q)onlyifthetwoonstanttermsin(4.11)with
k=0anel eahother. This meansthatthe
2n+1
(q) shouldsatisfytheequation
(q 1=2 ;q) n+1 (q 1=2 ;q) n 2n+1
(q)(q 1=2 ;q) n [1 q n+1=2 2n+1
(q)℄=0: (4.12)
Consequently,
2n+1
(q)=1 q n+1=2
and,therefore, by (4.11) we obtain
H
2n+2
(x;q)+(1 q n+1=2
)H
2n (x;q)
=( 1) n+1 (q 1=2 ;q) n+1 n+1 X k=0 q k(k 1=2) (q 1=2 ;q) k (
n+1
k q n k q ) ( x 2 ) k
=( 1) n (q 3=2 ;q) n x 2 n X m=0 q (m+1)(m+1=2) (q 3=2 ;q) m (
n+1
m+1
q
n
m+1 q ) ( x 2 ) m
=( 1) n q n+1=2 (q 3=2 ;q) n x 2 n X m=0 q m(m+1=2 (q 3=2 ;q) m n m q ( x 2 ) m =q n+1=2 xH 2n+1 (x;q);
(4.13)
uponusingthereadilyveriedrelation
n+1
m+1
q
n
m+1 q =q n m n m q (4.14)
for the q-binomial oeÆient
n
k
q
. From (4.10) and (4.13) it thus follows that the
q-polynomialsH
n
(x;q) satisfythethree-term reurrenerelation oftheform
H
n+1
(x;q)+(1 q n=2
)H
n 1
(x;q)=q n=2
xH
n
(x;q): (4.15)
An examinationofthereurrene relation(4.15) reveals that theH
n
(x;q) arerelatedto the
disrete q-Hermite polynomials ~
h (x;q) of type II. Indeed, hange the base q ! q 2
in(4.15)
and substituteintoit
H
n (x;q
2
)=q
n(n 1)=2
~
h
n
(x;q) (4.16)
toobtainthereurrenerelation(1.8)forthedisreteq-Hermitepolynomials ~
h
n
(x;q)oftype
II.
There are two immediate onsequenes of the formula (4.16). Firstly, ombining (4.16)
with(2.8), one obtainstherelationship
~
h
2n
(x;q)=( 1) n q n(2n 1) (q 2 ;q 2 ) n L ( 1=2) n (x 2 ;q 2 ); ~
h (x;q)=( 1) n q n(2n+1) (q 2 ;q 2
betweenthedisreteq-Hermitepolynomials ~
h
n
(x;q)oftypeIIandtheq-Laguerrepolynomials
(2.3)with=1=2. Seondly,oneanrepresenttheq-polynomialsH
n
(x;q),initiallydened
by(2.8), intheuniedform (f(1.7))
H
n (x;q
2
)=i n
2
0 q
n
;ix
|
q; q
n !
=q
n(n 1)=2
x n
2
1 q
n
;q 1 n
0
q
2
; q 2
=x 2
!
:
(4.18)
As a onsistenyhek, one may try to verifydiretly thatH
n (x;q
2
), denedby(4.18),
doesreallyoinidewiththeexpressions,given by(2.8)foreven andoddvaluesofn,
respe-tively. Thiswilllead to thetwo identities
2
1 q
n
;y
0
q;q
n+1=2 !
=q
n(n 1)=2
y n
2
1 q
n
;q 1=2 n
0
q;q=y
!
;
2
1 q
n
;y
0
q;q
n+3=2 !
=q
n(n+1)=2
y n
2
1 q
n
;q
(n+1=2)
0
q;q=y
!
;
(4.19)
betweenthe terminating basi hypergeometriseries
2
1
. The relations (4.19) are
straight-forwardto verifybyusingtheexpansion
(x;q)
k =
k
X
j=0 q
j(j 1)=2
k
j
q ( x)
j
(4.20)
ofthe q-shifted fatorial(x;q)
k
intermsof powersof thevariable x.
Yetanotheronsequeneoftherelation(4.16)anbeformulatedasthefollowingorollary
oftheorem (1)in setion3.
Corollary 2 Thesequeneof the disreteq-Hermite polynomials ~
h
n
(x;q) of typeII,dened
by (1.7), satises the orthogonality relation
1
Z
1 ~
h
m (x;q)
~
h
n (x;q)
dx
E
q 2(x
2
) =q
n 2
(q;q)
n (q;q
2
)
1=2 Æ
mn
(4.21)
on the whole real line R with respet to the ontinuous positive weight funtion w(x) =
1=E
q 2(x
2
).
Proof. Changethebase q!q 2
in(3.1) and thenusethe relation(4.16) to obtain(4.21).
Aswenotedintheintrodution,thedisreteq-Hermitepolynomials ~
h
n
(x;q)oftypeIIare
knownaspolynomialswiththedisreteorthogonality,supportedonthepointsx=q k
;>
0;k 2 Z. The fat that they satisfy also the ontinuous orthogonality relation (4.21) does
notontradit the general theory of orthogonal polynomials [22 , 23 ℄. The point is that the
Hamburgermoment problemassoiatedwithf ~
h
n
(x;q)gis indeterminate,andthereforethey
are orthogonal with respet to an innite lass of weight funtions, both ontinuous and
disrete ones [19 ℄, [24℄{[26 ℄. C.Berg studied in [27 ℄ some families of disrete solutions to
indeterminate moment problems and showed how they an be used to generate absolutely
ontinuoussolutionsto thesame momentproblems. Inpartiular,C.Bergderivedin[27 ℄the
ontinuousweightfuntionw(x)=1=E
q 2(x
2
)forthedisreteq-Hermitepolynomials ~
h
n (x;q)
of type II and evaluated its moments. It should be emphasized that in our approah the
orthogonalityrelation(4.21)emergesasasimpleonsequeneoftheontinuousorthogonality
relation(2.4) fortheq-Laguerre polynomialsL (1=2)
Observe that the ontinuous orthogonality relation (4.21) is useful for deriving some
integrals, involving the q-exponential funtion E
q
(z). We do not go into this question in
depth but merely present one of them as an example. One of the generating funtions for
thedisreteq-Hermite polynomialsoftypeIIhastheform (see [4 , p. 119℄)
1
X
n=0 q
n(n 1)=2
(q;q)
n ~
h
n (x;q)t
n
= E
q (xt)
E
q 2(t
2
)
: (4.22)
Multiplybothsidesof(4.21) byaonstantfator(itq m=2
) m
( iq n=2
) n
=(q;q)
m (q;q)
n
andsum
overindiesm andn fromzero toinnitywith theaidof thegeneratingfuntion(4.22) and
thedenitionof anotherJakson's q-exponentialfuntion
e
q (z):=
1
(z;q)
1 =
1
X
n=0 z
n
(q;q)
n
; jzj<1: (4.23)
Thisresultsin thefollowingintegral
1
Z
1 E
q (iq
1=2
xt)E
q ( iq
1=2
x)
E
q 2(x
2
)
dx=(q;q 2
)
1=2 e
q (t)E
q 2( qt
2
)E
q 2( q
2
); jtj<1:
(4.24)
Sine lim
q!1 e
q
((1 q)z) =e z
and theq-exponential funtionE
q
(z) has thesame limit (see
(3.7)), theformula(4.24) isa q-extensionof thewell-knownintegral
1
Z
1 e
2i(x y)t
e t
2
dt= p
e (x y)
2
;
whihreetstheimportantpropertyofthenormaldistributione x
2
ofbeingitsownFourier
transform.
5 Conluding remarks
We have disussedinthepreedingsetionshowoneanonstrut thepartiularpolynomial
sequene ofq-extensionsof theHermitepolynomials,eitherintermsof theq-Laguerre
poly-nomials L ()
n
(x;q), =1=2, orin terms of thedisrete q-Hermite polynomials ~
h
n
(x;q) of
type II. Thesequene so denedsatises theontinuousorthogonalityon R withrespet to
thepositive weight funtionw(x)=1=E
q (x
2
). It wasshownthat thisorthogonalityrelation
leadsto an interestingintegral,involvingJakson's q-exponentialfuntions.
It seems that the same approah an be implemented for deriving a q-extension of the
generalizedHermitepolynomialsH ()
n
(x)[28 ℄{[30℄withtheontinuousorthogonalityproperty
(theaseofdisreteorthogonalityrequiresadierenttehnique,see[31 ℄). Thisstudyisunder
way.
Aknowledgments
Wearegratefulto C.Bergfordisussionsandforhavingdrawnourattentionto the
refer-ene[27℄. TheresearhofRANhasbeenpartiallysupportedbytheMinisteriodeCieniasy
Tenologa ofSpain under thegrant BFM-2000-0206-C04- 02 , theJunta de Andaluaunder
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