On a q-extension of the Hermite polynomials Hn(x) with the continuous orthogonality property on R

Texto completo

(1)

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)

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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

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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)

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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 ;

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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

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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

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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

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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)

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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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Figure

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Referencias

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