The q-classical polynomials and the q-Askey and Nikiforov-Uvarov Tableaus

Texto completo

(1)

R.

Alvarez-Nodarse a;b 1

and J. C. Medem a 2

a

Departamento deAnalisisMatematio. UniversidaddeSevilla.

Apdo. 1160,E-41080Sevilla,Spain

b

InstitutoCarlosIdeFsiaTeoriayComputaional,UniversidaddeGranada,

E-18071Granada,Spain

January 8,2000

Abstrat

Inthis paperweontinue the study ofthe q lassial(disrete) polynomials (in theHahn's

sense)startedin [18℄. Herewewill ompareourshemewiththewellknownq AskeySheme

andtheNikiforov-UvarovTableau. Also,newfamiliesofq polynomialsareintrodued.

Introdution

The so-alled q polynomials onstitute a very important and interesting set of speial funtions

and more speiallyof orthogonal polynomials. They appear in several branhes of the natural

sienes, e.g., ontinuedfrations, Eulerianseries, theta funtions, elliptifuntions,...; see [3 , 9℄,

quantum groups and algebras [14 , 15 , 25 ℄, among others (see also [10 , 20 ℄). They have been

intensively studied in the last years by several people (see e.g. [13 ℄) using several tools. One of

them is the one reviewed in [13 ℄ whih is based on the basi hypergeometri series [10 ℄ and was

developed mainly by the Amerian Shool starting by the works of Andrews and Askey (see e.g.

[4 ℄,theliteratureon thismethodissovastthatwearenotabletoinludeithere,avery omplete

list is given in [13 ℄) and lead to the so-alled q Askey Tableau of hypergeometri polynomials

[13 ℄. In other diretion,the Russian (former Soviet) shool, starting from theworksby Nikiforov

and Uvarov [21 ℄ and further developed by Atakishiyev and Suslov (see e.g. [5 , 6, 20 , 23 , 24 ℄ and

referenes ontainedtherein), have onsidered thedierene analog innon-uniform latties of the

hypergeometri dierential equation [22 ℄, from where the hypergeometri representation of the

q polynomials follows in a very simple way [5, 23 ℄. This shema leads to the Nikiforov-Uvarov

tableau [20 , 23 ℄ for the polynomial solutions of the dierene hypergeometri equation on

non-uniform latties. A speial mention deserves the paper by Atakishiev and Suslov [6 ℄ where a

dierene analog of the well known method of undeterminated oeÆients have been developed

for the hypergeometri equation on non-uniform latties and also give a lassiation similar to

the Nikiforov and Uvarov 1991 one but forthe q speialfuntions (not only forthe polynomials

solutions).

Our main aims here are two: to ontinue thestudy started in[18℄using the algebrai theory

developedbyMaroni[16 ℄andtolassifytheq lassialpolynomialsandomparewiththeq Askey

and Nikiforov & Uvarov Tableaus. In fat, in [18 ℄ we have proven several haraterization of

the q lassial polynomials as well as a very simple omputational algorithm for nding their

main harateristis (e.g. the oeÆients of the three-term reurrent relation, struture relation

of Al-Salam Chihara, et). Going further, we will give here a \very natural" lassiation of the

q lassialpolynomials introdued by Hahn in his paper [11 ℄, i.e., we willlassify all orthogonal

polynomialsequenessuh that theirq dierenes,dened byf(x)=

f(qx) f(x)

(q 1)x

areorthogonal

in the widespread sene: the q Hahn Tableau (a rst step on this in the frame work of the

q Askey tableau was done in [15 ℄). Notie that the aforesaid polynomials are instanes of the

1

Phone: +34954557997,Fax: 954557972. E-mail: rania.es

2

(2)

q polynomialsonthelinearexponentiallattiex(s)=

1 q

s

. Forseveral surveysonthislattieand

their orresponding polynomials see [2 , 4, 5 , 8 , 10 , 13 , 20 , 24 ℄. (see also setion 3.2 from below).

Furthermore,wewillompareourlassiation(q HahnTableau)withtheaforesaidtwoShemas.

Fromthisomparationwendthattherearemissingfamiliesintheq AskeyShema(oneofthem

isa non-positivedenitefamily) andusingtheresultsof[18 ℄ westudythemwithdetails. Alsothe

orrespondeneofthisq HahnShemaandtheNikiforov&Uvarovonewillbestablished. Insuh

a way a omplete orrespondene between the q lassial familiesof the q Askey and Nikiforov

&UvarovTableaus forexponentiallinear lattieswillbe shown.

Thestrutureofthepaperisasfollows. InSetion1weintroduesomenotationsanddenitions

useful for the next ones. In Setion 2, the q weight funtions are introdued and omputed

for all q lassial families. This will allow to lassify all orthogonal polynomial families of the

q Hahntableau. Finally,inSetion3,severalappliationsareonsidered: thelassiationofthe

q lassialpolynomials (q Hahn Tableau), the integral representation for the orthogonality, the

hypergeometrirepresentationoftheseq lassialpolynomialsaswellasthedetailedstudyoftwo

new familiesof q polynomials.

1 Preliminaries

In this setion we will give a brief survey of the operational alulus and some basi onepts

and resultsneeded fortherest of thework.

Let P be the linear spae of polynomial funtions in C with omplex oeÆients and P

be

its algebrai dual spae, i.e., P

is the linear spae of all linear appliations u : P ! C. In the

followingwewillrefertotheelementsofP

asfuntionalsandwewilldenotethemwithboldletters

(u ; v;:::).

Sine the elements of P

are linear funtionals, it is possible to determine them from their

ations on a given basis (B

n )

n0

of P, e.g. the anonial basis of P, (x n

)

n0

. In general, we will

represent the ation of a funtional over a polynomial by formula hu;i; u 2 P

; 2 P, and

therefore a funtionalis ompletely determined by a sequene of omplex numbers hu;x n

i =u

n ,

n0,theso-alled momentsof thefuntional.

Denition 1.1 Let (P

n )

n0

be a basis sequene of P. We say that (P

n )

n0

is an orthogonal

polynomial sequene (OPS in short), if and only if there exists a funtional u 2 P

suh that

hu ;P

m P

n i = k

n Æ

mn , k

n

6=0; n 0, where Æ

mn

is the Kroneker delta. If k

n

>0 for all n 0,

we saythat (P

n )

n0

is a positive denite OPS.

Denition 1.2 Letu2P

bea funtional. Wesaythatu isa quasi-denitefuntionalifandonly

if there existsa polynomial sequene (P

n )

n0

, whih isorthogonal with respet tou. If (P

n )

n0 is

positivedenite, we say that u is a positive denite funtional.

Denition 1.3 Given a polynomialsequene(P

n )

n0

, we say that (P

n )

n0

isa moni orthogonal

polynomialsequene (MOPSin short) with respet tou, and we denote it by (P

n )

n0

=mopsu if

and only ifP

n

(x)=x n

+lowerdegree terms and hu ;P

m P

n i=k

n Æ

nm ; k

n

6=0; n0.

Alsothenext theorem willbeuseful

Theorem 1.1 (Favard Theorem [7℄) Let (P

n )

n0

be a moni polynomial basis sequene. Then,

(P

n )

n0

is an MOPS if and only if there exist two sequenes of omplex numbers (d

n )

n0 and

(g

n )

n1

, suh that g

n

6=0, n1 and

xP

n =P

n+1 +d

n P

n +g

n P

n 1 ; P

1

=0; P

0

=1; n0; (1.1)

whereP

1

(x)0 andP

0

(x)1. Moreover, thefuntional u withrespetto whihthe polynomials

(P

n )

n0

are orthogonal is positive denite if and only if (d

n )

n0

is a real sequene and g

n

>0 for

(3)

In thefollowing,wewilluse thenotation:

Denition 1.4 Let 2P and a2C, a6=0. We all the operator H

a

:P !P, H

a

(x)=(ax),

a dilation of ratio a2C nf0g.

ThisoperatorislinearonPandsatises H

a

()=H

a H

a

. Alsonotiethatforanyomplex

numbera6= 0, H

a H

a

1 = I, where I is the identity operator on P, i.e., for all a6= 0, H

a

has an

inverse operator. In the following we will omit any referene to q in the operators H

q

and their

inverse H

q

1. So, H:=H

q , H

1

:=H

q 1.

Next, we willdenetheso alledq derivative operator[11 ℄. We willsupposealso that jqj6=1

(although itispossibleto weak thisondition).

Denition 1.5 Let 2P and q 2C nf0g, jqj6=1. The q derivative operator , is the operator

:P!P, dened by

=

H

H x x =

H

(q 1)x :

Theq 1

derivative operator ?

, is the operator ?

:P!P dened by

?

= H

1

H 1

x x =

H 1

(q 1

1)x :

In this way, and ?

will denote the q derivative andq 1

derivative of , respetively.

The abovetwooperators and ?

arelinear operatorson P, and

x n

= Hx

n

x n

(q 1)x =

(q n

1)x n

(q 1)x

=[n℄x n 1

; n>0; 1=0; (1.2)

i.e., 2P. Here [n℄; n2N, denotesthebasiq numbern denedby

[n℄= q

n

1

q 1

=1+q+:::+q n 1

; n>0; [0℄=0: (1.3)

Also theq 1

numbers[n℄ ?

,dened by[n℄ ?

= q

n

1

q 1

1 =q

1 n

[n℄willbeused.

Notie that ?

is not the inverse of . In fat they are related by H ?

= ; H

1

=

?

.

The q derivativesatises theprodutrule()=+H=H +.

Denition 1.6 Let ! a derivable funtion atx=0 suh that 8a2dom!, aq2dom!. Then, we

will dene the q derivativeof ! by the expression

!=

H! !

Hx x =

H ! !

(q 1)x

; x6=0; !(0)=! 0

(0): (1.4)

Denition 1.7 Let u2P

and 2P. We denethe ationof adilation H

a

andthe q derivative

on P

by the expressions H

a :P

!P

, hH

a

u;i=hu ;H

a

i, :P

!P

, hu;i= hu;i,

respetively.

Denition 1.8 Let u2P

and 2P. Wedene a polynomial modiation of afuntional u, the

funtional u ,hu ;i=hu;i; 82P.

Notiethat we use thesame notationforthe operatorson P and P

. Whenever itis notspeied

on whih linearspae an operatorats,it willbe understoodthatit ats onthe polynomialspae

P.

Denition 1.9 Let u 2P

be a quasi-denitefuntional and (P

n )

n0

=mops(u ). We say that u

or (P

n )

n0

are q lassi funtionalor MOPS,respetively, ifandonlyif thesequene(P

n+1 )

n0

(4)

Notie that in the Hahn denition [11 ℄ q is a real parameter and here, in general, q 2 C nf0g,

jqj6=1.

In the following (Q

n )

n0

will denote the sequene of moni q derivatives of (P

n )

n0 , i.e.,

Q

n =

1

[n+1℄ P

n+1

,forall n0.

Theorem 1.2 (Medem et al. [17 , 18 ℄) Let u 2 P

bea quasi-denite funtional. and (P

n )

n0 =

mops(u). Then, the following statements are equivalent:

(a) u and (P

n )

n0

are,respetively, a q lassial funtional and a q lassial MOPS.

(b) Thereexists a pair of polynomials and , deg2, deg =1, suh that

(u)= u: (1.5)

() (P

n )

n0

satises the q SL dierene equation

?

P

n +

?

P

n =

b

n P

n

; n0; (1.6)

i.e.,P

n

are the eigenfuntionsof the Sturm-Liouvilleoperator ?

+ ?

orresponding to

the eigenvalues b

n .

Moreover, if

(x)=bax 2

+ax+a;_ (x)= b

bx+

b; b

b6=0; (1.7)

then, the quasi-denitenessof u implies [n℄ba+ b

b6=0 and the following equivalenes hold

[n℄ba+ b

b6=0; n0 () b

n 6=

b

m

; 8n;m1;n6=m () b

n

6=0;8n1:

Theorem 1.3 Letu2P

,beaquasi-denitefuntional,(P

n )

n0

=mopsuandQ (k)

n =

1

[n+1℄

(k )

k

P

n+k ,

where[n+1℄

(k)

[n+1℄[n+2℄:::[n+k 1℄. The following statements are equivalent:

(a)(P

n )

n0

is q lassial, (b)(Q (k)

n )

n0

is q lassial, k 1.

Moreover, if u satises the equation (u) = u , deg 2 and deg = 1, then (Q (k)

n ) is

orthogonal with respet to v (k)

=H (k)

u, H (k)

= Q

k

i=1 H

i 1

,and it satises

( (k)

v (k)

)= (k)

v (k)

; deg (k)

2 deg (k)

=1;

where (k)

=H k

and (k)

= +

P

k 1

i=0 H

i

,and theyarethe polynomial solutions of the q SL

equation

SL (k)

Q (k)

n =

(k)

?

Q (k)

n +

(k)

?

Q (k)

n =

b

(k)

n Q

(k)

n

; (1.8)

wherethe polynomials (k)

and (k)

and the eigenvalues b

(k)

n are

(k)

=q 2k

b ax

2

+q k

ax+

a; (k)

=([2k℄ba+ b

b)x+([k℄a+b); b

(k)

n

=[n℄ ?

([2k+n 1℄ba+ b

b): (1.9)

Furthermore,in[17 , 18℄the followingresultwasproven:

Theorem 1.4 Let u 2 P

, be a quasi-denite funtional, (P

n )

n0

= mopsu, ; ?

; 2 P, suh

that ?

=q 1

+(q 1

1)x , deg2,deg ?

2anddeg =1. Then,the followingstatements

are equivalent

(a) u and (P

n )

n0

=mopsu are q lassial and (u)= u,

(b) u and (P ) =mopsu are q 1

lassial and ?

( ?

(5)

() Thereexist apolynomial 2P, deg2andthreesequenes ofomplexnumbersa

n ; b

n ;

n ,

n

6=0, suh that

P

n =a

n P

n+1 +b

n P

n +

n P

n 1

; n1; (1.10)

(d) there exist a omplex numbers e

n ;h

n

, suh that

P

n =Q

n +e

n Q

n 1 +h

n Q

n 2

; n2: (1.11)

(e) There exist a polynomial 2P, deg 2 and a sequene of omplex numbers r

n , r

n 6=0,

n1 suh that

P

n u=r

n

n

(H (n)

u); H (n)

= n

Y

i=1 H

i 1

; r

n =q

( n

2 )

n

Y

i=1

[2n i 1℄ba+ b

b

1

; n1:

(1.12)

2 The q weight funtion !

2.1 Denition and rst properties

Inthissetionwewillonsidertheso-alledweightfuntionsforq lassialpolynomials. Thenext

propositionan beproven straightforward(see e.g. [12℄).

Proposition 2.1 Let ! a funtion suh that if a 2 dom!, aq 1

2 dom! and that satises the

dierene equation

?

(!)=q ! () !=qH( ?

!); ; 2P; ?

=q 1

+(q 1

1)x : (2.1)

Then, the following two equations are equivalent

?

P

n +

?

P

n =

b

n P

n

; ()

?

(!P

n )=q

b

n !P

n

; n1: (2.2)

Theabovepropositionallowsustogeneralizethelassialproeduretotheq aseforobtaining

almost all the harateristis of the MOPS. The equation (2.1) is usually alled the q Pearson

equation andits solution! isknownastheq weight funtionand itallowsto rewrite the

Sturm-Liuovilleequation (1.6) in its self-adjoint form (2.2). Moreover, theweight funtion! allow usto

obtainthe\standard" q Rodriguesformulaand also justify theq integralrepresentation forthe

orthogonalityrelation. In suh away itisnatural to give the following

Denition 2.1 Let u 2 P

, be a quasi-denite funtional satisfying the distributional equation

(1.5), where ; 2P,deg2, deg =1and (P

n )

n0

=mopsu . Wesay that ! isthe q weight

funtionassoiated to u (respetively to (P

n )

n0

) if! satises the equation (2.1) ?

(!)=q !.

The last denition allows us to rewrite the q SL equation (1.8) in its self-adjoint form. In

fat, an straightforwardalulationsshow that, if! (k)

satises theq Pearsonequation

?

( (k)

! (k)

)=q (k)

! (k)

; (2.3)

where (k)

and (k)

are given in(1.9), then(1.8) an be rewritteninits self-adjoint form

?

( (k)

! (k)

Q (k)

n )=q

b

(k)

n !

(k)

Q (k)

n

; n1; k =0;1;:::;n: (2.4)

Proposition 2.2 Let ! the solution of (2.1) and ! (k)

the solution of (2.3). Then,

! (k)

= (n 1)

! (n 1)

==H (n)

!; ! (0)

(6)

Proof: Westartfromtheq Pearsonequation(2.3)andrewriteitinitsequivalentform (k)

! (k)

=

qH[ (k)

℄ ?

H! (k)

,where[ (k)

℄ ?

=q 1

(k)

+(q 1

1)x (k)

= ?

,forall k2N. Thus,bysubstituting

! (k)

=H (n)

! in (k)

! (k)

=qH ?

H! (k)

,we nd

(k)

! (k)

=qH[ (k)

℄ ?

H! (k)

()H k

(H H k 1

!)=qH ?

HH k

H! ()

!=qH ?

H !() ?

(!)=q !;

from wherethepropositionfollows.

Remark 2.1 Notie that the polynomials ( (k)

) ?

and ( ?

) (k)

are very dierent. In fat, the rst

one together with (k)

are the orresponding polynomials that appear in the q 1

distributional

equation satised by the funtional v (k)

, i.e., the funtional with respet to whih the k th moni

derivatives Q (k)

n

areorthogonal, (see Proposition 1.4)

( (k)

v (k)

)= (k)

v (k)

()

?

( (k)

) ?

v (k)

= (k)

v (k)

; (

(k)

) ?

= ?

; 8k2N;

whereas the seond one joint with ( ?

) (k)

are the polynomial oeÆients of the q 1

SL equation

( ?

) (k)

?

Q ?(k)

n (

?

) (k)

=( b

?

) (k)

n Q

?(k)

n

, of the n th q 1

derivative Q ?(k)

n

of the polynomials P

n ,

Q ?(k)

n =

1

[n+1℄ ?

(k ) [

?

℄ n

P

n+k

or the q 1

distributional equation satised by the funtional v ?(k)

,

?

[( ?

) (k)

v ?(k)

℄=( ?

) (k)

v ?(k)

; (

?

) (k)

=H k

?

; 8k2N;

i.e., the funtional withrespet towhih the k th moni derivatives Q ?(k)

n

areorthogonal.

2.2 Computation of the q weight funtions

This setion is devoted to obtainthe q weight funtion assoiated to all q lassialfuntionals,

i.e.,thequasi-denitefuntionalsorrespondingto theMOPSinthewidespreadsense

u;P 2

n

6=0,

for all n 0. In fat, Theorem 2.1 and 2.2 will give, in a very natural way, the key for the

lassiation of allq lassialorthogonal polynomials.

In thefollowing we onsiderthe ase when jqj<1 (jq 1

j>1). Alsowe willuse the standard

notation (a;q)

n

=(1 a)(1 aq)(1 aq n 1

) forn 1, (a;q)

0

1 for theq analogue of the

Pohammer symbol, and (a;q)

1 =

Q

1

n=0

(1 aq n

), fortheabsolutelyonvergent innite produt

forjqj<1.

Firstofall,we willrewritetheq Pearsonequation(2.1)

?

(!)=q ! () !=qH ?

H! ()

?

!=q 1

H 1

H 1

!; (2.6)

and solve theresulting equation bythe reurrent proedure shown ingure1.

Figure 1. Reurrent shema usingtheq dilation.

w=H n

w qH

?

H

qH ?

:::H n 1

qH ?

| {z }

H (n)

qH ?

= Q

n 1

k=0 q

?

(q k +1

x)

k H

2

H 2

w=H 2

(qH ?

)H 3

w

::::::

1

R

1

R

HHw=H(qH ?

)H 2

w

? H

w=H 2

w qH

?

H

qH ?

w=qH

?

Hw

-? H

w=Hw qH

?

(7)

In the ase when ! is ontinuous at 0 and !(0) 6= 0, taking the limitn ! 1, we nd, sine

lim

n!1 H

n

w=lim

n!1 w(q

n

x)=w(0),

!=!(0)limH (1)

qH ?

=!(0) lim

n!1 H

(n) qH

?

=!(0) 1

Y

n=0 qH

?

: (2.7)

ThenextstepistoobtainanexpliitexpressionfortheprodutH (1)

qH ?

. Fordoingthatweneed

a lemmawhihis interestinginits ownright.

Lemma 2.1 If is an n th degree polynomial with an independent term (0) = 1, and zeros

a

i

2Cnf 0g , i=1;2;:::;n,then

H (1)

=(a 1

1 x;q)

1 (a

1

2 x;q)

1 (a

1

n x;q)

1 :=(a

1

1 x;a

1

2

x;;a 1

n x;q)

1 ;

is an entire funtion of x with zeros at a

i q

k

, i = 1;2;:::;n and k 0. Furthermore, if = is

a rational funtion suh that (0)= (0) 6=0 and with non-vanishing zeros of its numerator and

denominator, then,

H (1)

=

(a 1

1 x;q)

1 (a

1

2 x;q)

1 (a

1

n x;q)

1

(b 1

1 x;q)

1 (b

1

2 x;q)

1 (b

1

m x;q)

1 =

(a 1

1 x;a

1

2

x;;a 1

n x;q)

1

(b 1

1 x;b

1

2

x;;b 1

m x;q)

1 ;

it is a meromorphi funtion with zeros at a

i q

k

, i = 1;2;:::;n and k 0 and poles at b

j q

l

,

j =1;2;:::;m and l0, where a

i

2C, i=1;2;:::;n and b

k

2C, k =1;2;:::;m, are the zeros

of the numerator and denominator of =, respetively.

Proof: The proof is based on the fat that, if is a polynomial of degree n with non vanishing

zeros and (0)=1, thenitadmits thefatorization

=A(x a

1

)(x a

2

)(x a

n

)=( 1) n

Aa

1 a

2 a

n

| {z }

(0)=1

(1 a 1

1

x)(1 a 1

2

x)(1 a 1

n x):

Then, H (k)

=(a 1

1 x;a

1

2

x;;a 1

n x;q)

k

and so, H (1)

=(a 1

1 x;a

1

2

x;;a 1

n x;q)

1

. This

fun-tion is an entire funtion due to the Weierstrass Theorem (see e.g. [1, x4.3℄). The proof of the

seondstatementisanalogousandthefuntionH (1)

ismeromorphibeauseisaquotientoftwo

entirefuntions(see e.g. [1 , x4.3℄).

Now, if(0)6=0,theabovelemma leadsusto thefollowingwellknownresult [11 ℄

Theorem 2.1 Let(P

n )

n0

=mopsusatisfyingtheq Sturm-Liouvilleequation(1.6). Ifwedenote

by a

1 and a

2

the zeros of andby a ?

1 and a

?

2

the zerosof ?

(seeProposition1.4), and all they are

dierent from 0, then the following expressions for the q weight funtions ! hold

?

q weight funtion!(x)

b a ?

(x a ?

1

)(x a ?

2 ), ba

?

a ?

1 a

?

2

6=0 !(x)= (a

?

1 1

qx;a ?

2 1

qx;q)

1

(a 1

1 x;a

1

2 x;q)

1

b a(x a

1

)(x a

2 ), baa

1 a

2

6=0 a

?

(x a ?

1 ), a

?

a ?

1

6=0 !(x)=

(a ?

1 1

qx;q)

1

(a 1

1 x;a

1

2 x;q)

1

_ a ?

6=0 !(x)=

1

(a 1

1 x;a

1

2 x;q)

1

a(x a

1 ), aa

1

6=0 !(x)=

(a ?

1 1

qx;a ?

2 1

qx;q)

1

(a 1

1 x;q)

1

b a(x a

1

)(x a

2 ), baa

1 a

2 6=0

_

a6=0 !(x)=(a

?

1 1

qx;a ?

2 1

qx;q)

(8)

Proof: Sine (x) = ba(x a

1

)(x a

2

) and ?

= q 1

+(q 1

1)x = ba ?

(x a ?

1

)(x a ?

2 ), we

have (qH ?

)(0)=q ?

(0)=(0),sothepolynomialsqH ?

andhave thesameindependentterm.

Usingthepowerexpansionof thepolynomialsand ?

(x)=bax 2

+ax+a;_ ?

(x)=ba ?

x 2

+a ?

x+a_ ?

;

wehaveba ?

=q 1

b a+(q

1

1) b

b,a ?

=q 1

a+(q

1

1)

banda_ ?

=q 1

_

a,where, b

b;

baretheoeÆient

of thepowerexpansionof (see Eq. (1.7)). Thus,

8

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

:

deg<2=)ba=0=)ba ?

6=0=)deg ?

=2;

deg=2=)ba6=0 8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

: b

b6= b a

1 q =)ba

?

6=0=)deg ?

=2;

b

b= b a

1 q =)ba

?

=0 8

>

>

>

<

>

>

>

:

b6= a

1 q

=)deg ?

=1;

b= a

1 q

=)deg ?

=1:

In all ases we an apply diretly the above lemma whih immediately leads us to the desired

result. Notie also that all the obtained funtions are meromorphi and so, they are ontinuous

and non-vanishingat x=0, sowe an supposewithoutanylossof generalitythat !(0)=1.

In the ase when (0) = 0, it is easy to see that ?

(0) =0. This ase requires a more detail

study. In the following we should keep in mind that for the quasi-deniteness of u 6 0 and

and shouldbe oprimepolynomials (see [18℄).

Proposition 2.3 Letubeaq lassialfuntional satisfyingthedistributionalequation(1.5)with

=ba x 2

+ax, jbaj+j aj>0, and = b

bx+

b, b

b6=0. Thenthe following ases, ompatible withthe

quasi-deniteness of u, appear:

(a) If =bax 2

, ba6=0, then, deg ?

=2 and its two zeros aredierent, or deg ?

=1.

(b) If =bax 2

+ax, baa6=0, then, deg ?

=2, or deg ?

=1.

() If =ax, a6=0, then, deg ?

=2.

Proof:

(a) Sine =bax 2

, then = b

bx+

b, with

b6=0, otherwise divides . Therefore, ?

=(q 1

b a+

(q 1

1) b

b )x 2

+(q 1

1)

b x has a non-vanishing oeÆient on x. If b

b 6= b a

1 q

then ba ?

6= 0 and

deg ?

=2 and ?

has two dierent zeros one of whih is loated at theorigin. If b

b = b a

1 q then

deg ?

=1.

Theother two ases areproven analogously.

Thenext step isto ndtheq weight funtionsforall possibleases aordingwith theabove

proposition(rememberthat (0)=0= ?

(0)). There aretwo largelasses. ClassIorresponding

to the asewhen and ?

have non-vanishingtermon x and IIwhen they have a vanishingterm

on x.

I. Westartwiththeasewhen and ?

havenot-vanishingtermonx. Inthisasethere arethree

dierent possibilities(sublasses):

(a) (x)=bax(x a

1 ),baa

1

6=0 and ?

(x)=ba ?

x(x a ?

1 ),ba

?

a ?

1 6=0,

(b) (x)=bax(x a

1 ),baa

1

6=0 and ?

(x)=a ?

x,a ?

6=0,

() (x)=ax,a6=0 and ?

(x)=ba ?

x(x a ?

),ba ?

a ?

(9)

To nd the orresponding q weight funtions we will rewrite the quotient qH ?

= = xq

(H ?

)

0 =x

0

, where = x

0

and H ?

= x(H ?

)

0

. In general,

0

(0) 6= (H ?

)

0

(0), so, in order

to apply a method, similar to the one used to prove Theorem 2.1, we willassume that ! an be

rewritten on the form ! = jxj

!

0

, 2 Cn f0 g, where is a free parameter to be found. An

straightforward alulations show that if ! satises a q Pearson equation (2.6) then !

0

satises

theequation

0 !

0

=aqH!

0 (H

?

)

0

,wherea=q

. So,

!

0 =H

n

(!

0 )

aq(H ?

)

0

0

; a=q

; or =Log

q (a);

whereLog

q

denotestheprinipallogarithmonthebasisq,jqj<1. Inthefollowing,wewillusethe

notation a= baa

1 and a

?

= ba ?

a ?

1

. Notie that, with this notation, =bax(x a

1 ) =bax

2

+ax

and ?

=ba ?

x(x a ?

1 )=ba

?

x 2

+a ?

x.

(a)In thisase,

aq(H ?

)

0

0 =

aq 2

b a ?

a ?

1 (a

?

1 1

qx 1)

b aa

1 (a

1

1

x 1) =

aq 2

a ?

(1 a ?

1 1

qx)

a(1 a

1

1 x)

:

Ifwehoosenow,a suhthat aq(H ?

)

0

(0)=

0

(0), i.e.,aq 2

a ?

=a,orequivalently,=Log

q (a)=

2+Log

q a

a ?

, we an apply the Lemma 2.1 to get, !

0 = !

0 (0)

(a ?

1 1

qx;q)

1

(a 1

1 x;q)

1

, whih leads, without

anylossofgenerality,to thefollowingweight funtion(herewe supposethat!

0

isontinuousand

!

0

(0)6=0)

!(x)=jxj

(a ?

1 1

qx;q)

1

(a 1

1 x;q)

1

; =Log

q

(a)= 2+Log

q a

a ?

: (2.8)

(b)In thisase,

aq(H ?

)

0

0 =

aq 2

a ?

a(1 a

1

1 x)

=) !

0 =!

0 (0)

1

(a 1

1 x;q)

1

; =Log

q

(a)= 2+Log

q

a

a ?

;

so,

!(x)= jxj

(a 1

1 x;q)

1

; =Log

q

(a)= 2+Log

q

a

a ?

:

Finally,inthelastase (), we obtain

!(x)=jxj

(a ?

1 1

qx;q)

1

; =Log

q

(a)= 2+Log

q

a

a ?

:

II.Let onsiderthe other ase, i.e., whenand ?

have a vanishingtermon x. In thisasethere

aretwo possibilities:

(i) deg 6= deg ?

whih is divided in two subases (a) = bax 2

, ?

= a ?

x, and (b) = ax,

?

=ba ?

x 2

,and

(ii) deg=deg ?

,whih also isdividedintwo subases (a) =bax 2

, ?

=ba ?

x(x a ?

1 ), a

?

1 6=0,

and (b)=bax(x a

1 ),

?

=ba ?

x 2

,a

1 6=0.

In bothases, themethod usedinthe aseIof non-vanishingoeÆientsan notbe used.

(i) Inorder to solvethe problemforaseII(i)we willgeneralizeanidea byHaker [12 ℄. Letus

denethefuntionh ()

: [0;1)!R denedby

h ()

(x)= p

x log

q x

; 6=0;

whih hasthefollowingpropertyH h

=x

h

,or, equivalently,h

(qx)=x

h

(x), forall x0.

If we nowdenethefuntion !=x

h (1)

,then, fortheaseII(i)a we have

H!=Hx

h (1)

=q

x

xh (1)

=q

x! =) xH!=q

x 2

(10)

then, omparingthisresultingequationwiththeq Pearsonequation(2.6) forthishoieof and

?

,bax 2

! =qa ?

xH!,we dedue thatthefuntion

!(x)=jxj

p

x log

q x 1

; = 2+Log

q b a

a ?

; x0; (2.9)

is thethesolutionof theq Pearsonequation and so,the orresponding q weight funtion.

Forthe aseII(i)b we have, inan analogous way,a similarsolutionbut involving thefuntion

h ( 1)

:

!(x)=jxj

q

x log

q 1

x +1

; = 3+Log

q a

b a ?

; x0: (2.10)

(ii)Inthisasethemethoddevelopedfortheaboveasesdoesnotwork. Infat,ifwetrytouse

themethodfortheaseI,aftersomestraightforwardalulations,wearrivetoaninnitedivergent

produt. For this reason we will solve the q Pearson equation using the equivalent equation

(2.6) in q 1

dilation q 1

H 1

H 1

! = ?

! (2.6), i.e., using a shema similarto the one given in

gure 1 but when the reurrene is solved in the \opposite" diretion to obtain the expression

! = H n

!H ( n)

q 1

H 1

?

, whih leads to the solution, by taking the limit n ! 1, ifthere exists

the value H 1

! =!(1). In suh a way,we have forthease II(ii)atheexpression

!=jxj

!

0

; 0 ; ?

=x ?

0 =x(ba

?

x+a ?

) ; H 1

=x(H 1

)

0 =x(q

2

b ax):

hene, the q 1

Pearsonequation takestheform

x ?

0 jxj

!

0 =q

1

x(H 1

)

0 H

1

(jxj

!

0

) =) ?

0 !

0 =q

1

(H 1

)

0 q

H 1

!

0 ;

and its solutionis

!

0

(0)=H n

w ?

0 H

( n) q

q 1

(H 1

)

0

?

0

a:=q

= H

n

!

0 H

( n) a

1

q 3

b ax

ba ?

x+a ?

:

Now, hoosing thevalue ,insuh a way thata 1

q 3

b a=ba

?

,i.e., = 3+Log

q ba

b a ?

we nd,

!

0

(0)=H n

w ?

0 H

( n) b a ?

x

b a ?

x+a ?

= H n

!

0 H

( n)

1

a ?

b a ?

x+a ?

=H n

!

0 n

Y

i=0

1

a ?

q i

b a ?

x+a ?

q i

:

Obviouslythe above produtis uniformlyonvergent inanyompatsubsetof theomplex plane

that not ontains the points fa ?

1 q

n

;n 0g S

f0g, where a ?

1 = a

?

=ba ?

is the non-vanishing zero

of ?

(in x = 0 the produt diverges to zero). Furthermore, this produt onverges at 1, so

!(1)=6=0,and thus

!(x)=jxj

w ?

0 =jxj

1

Y

n=0

1

a ?

q n

b a ?

x+a ?

q n

=jxj

H ( 1)

ba ?

x

b a ?

x+a ?

=jxj

1

( a ?

b a ?

x ;q)

1 ;

where = Log

q b a q

3

ba ?

, and whih,without any lossof generality, leads to the followingexpression

forthe q weight funtion

!(x)=jxj

1

(a ?

1 =x;q)

1 =jxj

e

q (a

?

1

=x); a ?

1 =

a ?

b a ?

x

; = 3+Log

q ba

b a ?

; (2.11)

where e

q

denotestheq exponentialfuntion[10 ℄.

AsimilarsituationhappensintheII(ii)bsubase. Inthisase, wehave

!

0

(x)=H n

!

0 H

( n) q

q 1

(H 1

)

0

?

a:=q

= H

n

!

0 H

( n) a

1

q 1

q 1

b a(q

1

x a

1 )

b a ?

x

(11)

Ifwenow hoose a 1

q 3

b a=ba

?

,we nd

!

0

(x)=H n

!

0 H

( n) b a ?

x a 1

q 2

b a a

1

b a ?

x

=H n

!

0 H

( n)

1 b a ?

qa

1

ba ?

x

=H n

!

0 H

( n)

1 a

1 q

x

;

whihisanabsoluteanduniformlyonvergentprodutinCnf 0g . Finally,sine!(1)=6=0,and

withoutanylossofgeneralitywe ndthefollowingexpressionfortheq weight funtion!

!(x)=jxj

(a

1 q=x;q)

1

; = 3 Log

q ba

b a ?

;

where a

1

is the non-vanishing zero of . All the above alulations an be summarize in the

followingtheorem:

Theorem 2.2 Let(P

n )

n0

=mopsusatisfyingtheq Sturm-Liouvilleequation(1.6). Ifwedenote

by a

1 anda

2

the zerosof andby a ?

1 anda

?

2

the zeros of ?

(seeProposition 1.4), andoneof them

are equal to 0, then the following expressions for the q weightfuntions ! hold

Case

?

!(x)

II(ii)a ba

?

x(x a ?

1 ); ba

?

a ?

1

6=0 jxj

1

(a ?

1 =x;q)

1

; =Log

q b aq

3

ba ?

b a x

2

, ba6=0

II(i)a a

?

x, a ?

6=0 jxj

p

x log

q x 1

; =Log

q b aq

2

a ?

I(a) ba

?

x(x a ?

1 ), ba

?

a ?

1

6=0 jxj

(a ? 1

1

qx;q)

1

(a 1

1 x;q)

1

, =Log

q aq

2

a ?

I(b) bax(x a

1 ), ba a

1

6=0 a

?

x, a ?

6=0 jxj

1

(a 1

1 x;q)

1

, =Log

q aq

2

a ?

II(ii)b ba

?

x 2

, ba ?

6=0 jxj

(a

1 q=x;q)

1

, = Log

q b a q

3

b a ?

I() ba

?

x(x a ?

1 ), ba

?

a ?

1

6=0 jxj

(a ? 1

1

qx;q)

1

, =Log

q aq

2

a ?

ax, a6=0

II(i)b ba

?

x 2

, ba ?

6=0 jxj

p

x log

q 1

x +1

, =Log

q aq

3

ba ?

3 Appliations

Inthissetionwewillonsidersomeappliationsoftheabovetheorems. Infatwewillshowhowthe

q weight funtions an be used to give an integral representation for theorthogonality. Another

interesting appliation is the already mentioned lassiation of all orthogonal families in the

q HahnTableau(in[23 ℄theorthogonalitywasnotonsidered). InfatTheorems2.1and2.2gives

anaturallassiation oftheq lassialorthogonalpolynomials. Alsobyusingtheq weightsone

an obtain anexpliit formula of thepolynomialssatisfying a Rodrigues-type formulain termsof

thepolynomialsoeÆientsand ?

from wherethehypergeometrirepresentationeasilyfollows.

The last have been done independently in [23 ℄ and [6 ℄ (see also [5 , 20 ℄) in the framework of the

dierene equations of hypergeometri type on the non-uniform latties. Here we willshow how

all the q lassial families an be obtained by ertain limiting proesses from the most general

aseof; Jaobi/Jaobifamily. Finally,wewillomparetheNikiforov&Uvarovandtheq Askey

Tableauswithourq HahnTableauandompletetheq Askeyonewithnewfamiliesoforthogonal

(12)

3.1 The q integral representation for the orthogonality

Inthissetionwewillshowhowtheq weightfuntionsandtheq SLequationleadtoaq integral

representation for the orthogonality. The tehnique used here is very ommon in the theory of

orthogonal polynomials(see e.g. [7, 13 ,20 ℄).

Firstof allwe introduethe q integralofJakson [10 , 25 ℄. ThisintegralisaRiemannsum on

an innitepartition faq n

;n0g,

Z

a

0

f(x)d

q

x=(1 q)a 1

X

n=0 f(aq

n

)q n

; and Z

b

a

f(x)d

q x=

Z

b

0

f(x)d

q x

Z

a

0

f(x)d

q x;

so,itisvalidthe q analogueoftheBarrowrule(hereF(x)isontinuousatx=0): Z

b

a

F(x)d

q x=

F(b) F(a),and therulesof integration byparts

Z

b

a

f(x)g(x)d

q x=H

1

f(x)g(x)

b

a q

Z

b

a

g(x) ?

f(x)d

q x;

Z

b

a

f(x)g(x)d

q

x=fg

b

a Z

b

a

H g(x)f(x)d

q x:

Obviouslyinthe above expressionsit isassumed thatthe funtionf is denedinthe

orrespond-ing partition's points. This Jakson q integral an be easily generalized to unbounded intervals

and unbounded funtions in a similar way as the Riemann integral [10 , 25 ℄. Furthermore, the

Riemann-Stieltjesdisreteintegralsrelatedwiththeq lassialpolynomialsan be representedas

q integrals (see e.g. [17 ,19 ℄).

Proposition 3.1 Let!beontinuousfuntioninx=0satisfyingtheq Pearsonequation ?

(!)=

q !, equivalentto the distributional equation(u)= u and let a;b omplex numbers suh that

the boundary ondition ?

!

b

a

=0, or equivalentlyH 1

!

b

a

=0 (w =qH( ?

w)) holds. Then,

Z

b

a P

n (x)P

m

(x)!(x)d

q

x=0; 8n6=m; (P

n )

n0

=mopsu:

Proof: The proof isstraightforward. We startfrom the self-adjoint form of theq SL equations

forthe polynomialP

n and P

m

,respetively:

[H 1

(!) ?

P

n ℄=

b

n !P

n

; [H

1

(!) ?

P

m ℄=

b

m !P

m :

If we multiply therst one by P

m

,the seond one byP

n

,takesthe q integralover (a;b) and use

theintegration bypart ruleswe nd

( b

n b

m )

Z

b

a !P

n P

m d

q x=

Z

b

a (!

b

n P

n )P

m d

q x

Z

b

a (!

b

m P

m )P

n d

q x=

= Z

b

a

H 1

(!) ?

P

n

P

m d

q x

Z

b

a

H 1

(!) ?

P

n

P

n d

q x=

=H 1

(!)W ?

q [P

m ;P

n ℄

b

a +

Z

b

a h

H

H 1

(!) ?

P

m

P

n

H(H 1

(!) ?

P

n

P

m i

d

q x;

where W ?

q [P

m ;P

n ℄ = P

m

?

P

n P

n

?

P

m

is the q Wronskian. The rst term in the last

equa-tion vanishsine the boundary onditions. The seond also vanishsine H

H 1

(w) ?

P

m

P

n

=wP P . Theresult followsfrom thefat thatforall n6=m, b

6= b

(13)

Remark 3.1 Notie that the hoie of the integration interval (a;b) is onditioned to guarantee

that R

b

a P

2

n !d

q

x 6= 0, n 0, for whih, it is enough that ! be ontinuous funtion and does not

vanishinsidethe intervalof integration. Thishasa diÆultysine,evenin thesimplest ases, i.e.,

; families, ! has innite zeros a ?

i q

n

, n 1, and innite poles, a

i q

n

, n 0. Notie also that

natural values for (a;b) are the roots of ?

or the roots of (q 1

x).

Of speialinterestis the studyof thepositive denite ase, i.e., the ase when R

b

a P

2

n !d

q x >0

foralln0. FordoingthatweanusetheFavardtheorem. Thedetailedstudyofpositivedenite

ase willbe onsideredina forthomingpaper.

3.2 Classiation of the q lassial polynomials

Sine the equation (1.5) (and so the Sturm-Liouville equation (2.2)) gives all the information

about the q lassialfuntional (and then about the orresponding MOPS), it is natural to use

them for lassifying the q lassial polynomials. Moreover, all this information is ondensed in

the polynomials and ?

instead of and (and more exatly in their zeros) as it is shown

in Theorems 2.1 and 2.2. So it is natural to use the zeros of and ?

to lasify all families of

q lassialorthogonalpolynomials[17,19 ℄.

Insuhaway,sine(0)=0ifand onlyif ?

(0)=0,itisnatural,inarststep,tolassifythe

q lassialpolynomials into two widegroups: the ; families, i.e., thefamiliessuhthat (0)6=0

and the 0 families, i.e., the ones with(0) = 0. Next, we lassify eah member inthe aforesaid

two wide lasses interms of the degree of the polynomials and ?

as wellas themultipliityof

theirroots inthe aseof 0 families. In fat, ifhas two simpleroots, thepolynomials belong to

the 0 Jaobi/|family whileifthe roots aremultiple, thenthey are 0 Bessel/| family. So, we

have thefollowingsheme fortheq lassialOPS:

; families 8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

; Jaobi/Jaobi

; Jaobi/Laguerre

; Jaobi/Hermite

; Laguerre/Jaobi

; Hermite/Jaobi

0 families 8

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

:

0 Bessel/Jaobi

0 Bessel/Laguerre

0 Jaobi/Jaobi

0 Jaobi/Laguerre

0 Jaobi/Bessel

0 Laguerre/Jaobi

0 Laguerre/Bessel

Notiethatinthisshemeannotappearthefamilies; Laguerre/Laguerre,;

Laguerre/Her-mite ; Hermite/Laguerre and ; Hermite/Hermite sine the onnetion between and ?

, as

well as the 0 Bessel/Bessel ase sine they do not orrespond to a quasi-denite funtional(see

Proposition2.3).

3.2.1 Connetion with the Nikiforov-Uvarov and the q Askey Tableaus

Here we will identify our lassiation (sheme)of the q lassial polynomialswith the two well

known shemesbyNikiforovand Uvarov [23 ℄ andthe q AskeyTableau [13 ℄.

Westartwiththerstone. TheNikiforov-UvarovTableauisbasedonthepolynomialsolutions

oftheseondorderlineardiereneequationofhypergeometritypeinthenon-uniformlattiex(s):

~ (x(s))

4

4x(s 1

2 )

5y

n [x(s)℄

5x(s) +

~ (x(s))

2

4y

n [x(s)℄

4x(s) +

5y

n [x(s)℄

5x(s)

+y

n

[x(s)℄=0;

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

n

[x(s)℄2P[x(s)℄

x(s)= (q)q s

+ (q)q s

+ (q);q2C;

(14)

where (x)~ and ~(x) arepolynomials in x(s)of degree at most 2 and 1, respetively,and

n is a

onstant, or,written inits equivalentform

(s) 4

4x(s 1

2 )

5y

n [x(s)℄

5x(s)

+(s) 4y

n [x(s)℄

4x(s) +

n y

n

[x(s)℄=0;

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

2 ~

(x(s))4x(s 1

2

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

(3.2)

Here P[x(s)℄ denotes thelinear spae of polynomialsin x(s). Notie that, ifx(s)=

1 q

s

x,i.e.,

weare intheso-alledlinear exponential lattie, then

4y

n [x(s)℄

4x(s)

=y

n

(x) and 5y

n [x(s)℄

5x(s) =

?

y

n

(x); y

n

(x)y

n [x(s)℄:

Thus,usingthefatthat4x(s 1

2 )=q

1

2

4x(s), thehypergeometriequation(3.2) inthelinear

lattie x(s)=

1 q

s

an berewrittenas

(s) ?

y

n (x)+q

1

2

(s)y

n

(x)=

n q

1

2

y

n

(x); y

n

(x)2P;

from where, and usingthe identity=x(q 1) ?

+ ?

we arrive to theequation

[+q 1

2

(s)x(q 1)℄ ?

y

n (x)+q

1

2

(s) ?

y

n

(x)=

n q

1

2

y

n (x);

whihis nothingelsethat theq SLequation (1.6) where

(s)=+x(1 q) =q ?

; (s)=q 1

2

;

n = q

1

2 b

n

: (3.3)

Inotherword,theq SLequation(1.6)isaseondorderlineardiereneequationofhypergeometri

type in thelinear exponential lattie x(s)=

1 q

s

. The above onnetion allows us to identify all

the q lassial orthogonal polynomials(in thewidespreadHahn's sense) withthe q polynomials

intheexponentiallattie intheNikiforovetal. approah. Infat, usingtheexpliitexpressionof

thepolynomials(s) and (s)+(s)4(x 1

2

) intheexponentiallattie [23,Eqs. (84)-(85) page

241 and Table page 244℄, we an identify our 12 lasses of q polynomials with the ones given in

[23 ℄ (see Table 3.2.1).

In order to identify the q lassial polynomials with the ones given in the q Askey tableau

[13 ℄ werewrite theq SLequation (1.6) inthefollowingform:

HP

n

(+q 2

?

)P

n +q

2

?

H 1

P

n

=(q 1) 2

x 2

n P

n :

Then,asimpleomparisonoftheabovediereneequationwiththosegivenintheq AskeyTableau

allowsusto identifysomeofthefamiliesoftheq polynomialsgivenin[13 ℄withtheorresponding

q lassial ones, and so, with the ones in the Nikiforov-Uvarov Tableau. This will be given in

Table3.2.1.

From the above table 3.2.1 we see that the 0 Jaobi/Bessel and 0 Laguerre/Bessel families

lead to new families of orthogonal polynomials. The reason for that they do not appear in the

q Askey tableauwill be onsideredlatter on. Notie also that the lass N o

8 from the

Nikiforov-Uvarov tableau [23 , page 244℄ do not lead to any orthogonal polynomial sequene even in the

widespreadsenseonsideredhere.

3.3 The Rodrigues formula and hypergeometri representation

For thesake of ompletenesswe willinlude herethe identiation ofthe q lassialpolynomials

interms of thebasihypergeometriseries[10℄denedby

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

( 1) k

q k (k 1)

2

p r+1

; (3.4)

where, asbefore,(a;q)

k =

Q

k 1

(1 aq m

(15)

Table 3.2.1: Comparison of the Nikiforov-Uvarov, the q Askey and the q lassial polynomial

Tableaus

q lassialfamily () Nikiforov-UvarovTableau [23℄ =) q AskeyTableau [13 ℄

; Jaobi/Jaobi () Eq. (86)page242 =) The Big q Jaobi

q Hahn

; Jaobi/Laguerre () N o

6[23 , page 244℄ =) q Meixner

Quantum q Kravhuk

; Jaobi/Hermite () N o

12 [23 , page244℄ =) Al-Salam-CarlitzII

Disrete q 1

HermiteII

; Laguerre/Jaobi () N o

1[23 , page 244℄ =) Big q Laguerre

AÆne q Kravhuk

; Hermite/Jaobi () N o

2[23 , page 244℄ =) Al-Salam-CarlitzI

Disrete q Hermite

0 Bessel/Jaobi () N

o

4[23 , page 244℄ =) Alternativeq Charlier

0 Bessel/Laguerre () N o

11 [23 , page244℄ =) Stieltjes-Wigert

0 Jaobi/Jaobi () N

o

3[23 , page 244℄ =) The Little q Jaobi

q Kravhuk

0 Jaobi/Laguerre () N o

10 [23 , page244℄ =) q Laguerre

q Charlier

0 Jaobi/Bessel () N

o

7[23 , page 244℄ =) new OPfamily

0 Laguerre/Jaobi () N o

5[23 , page 244℄ =) Littleq Laguerre(Wall)

0 Laguerre/Bessel () N o

9[23 , page 244℄ =) new OPfamily

| N

o

8[23 , page 244℄ |

3.3.1 The Rodrigues formula

Letusrst obtainthe\standard" Rodriguesformula.

Proposition 3.2 Let u, u2P

bea q lassial quasi-denite funtional, (P

n )

n0

=mopsu, and

! the q weight funtiondened by the q Pearson equation (2.1). Then,

P

n =q

n

r

n

? n

(H (n)

!)

!

: (3.5)

Proof: Theproofofthispropositionisstraightforward. Infat,usingthedenitionofthe! (k)

we

obtain! (k)

= (k 1)

! (k 1)

,thus, usingtheequation(2.4) we have, forall n1,

?n

(H (n)

!)= ?n

(! (n)

Q (n)

0 )=

1

[1℄

?n 1

[ ?

( (n 1)

! (n 1)

Q (n 1)

1 ℄

(2:4)

=

= q

b

(n 1)

1

[1℄

?n 1

[! (n 1)

Q (n 1)

1

℄==q n

b

(n 1)

1

b

n

[1℄[n℄ !P

n :

Finally,usingtheexpliitexpressionforthe oeÆient r

n

(16)

TheRodrigues formula isvery usefulforndingtheexpliitexpressionof thepolynomialsP

n .

In fat,usingthe formula

?n

f(x)= q n 2 +n (1 q) n x n n X k=0 ( 1) k q k (k +1)

2 nk n k q f(q k n x); n k q = (q;q) n (q;q) k (q;q) n k ; where n 2 = n(n 1) 2

,one easilyobtains

P n = q n 2 r n (1 q) n x n n X k=0 ( 1) k q k (k +1)

2 nk n k q H (n) (xq k n )!(q n k x) !(x) ;

or, equivalently,

P n = r n ( 1) n (1 q) n x n n X k=0 ( 1) k q k 2 n k q H (n) (xq k )!(q k x) !(x) :

Now, taking into aount theq Pearsonequation(2.6)

H ! ! = qH ? () H 1 ! ! = q ? H 1 ;

weobtainthefollowingexpliitexpressionfortheq lassialpolynomials intermsof the

polyno-mialsand ? : P n = r n ( 1) n (1 q) n x n n X k=0 ( 1) k q k 2 +k n k q k 1 Y i=0 ? (xq i ) n k 1

Y

i=0 (xq

i

): (3.6)

This formula isequivalent to theone obtained in[5 ,Eq. (4.14)℄, [23 ,Eq. (33)℄ and [2, Eq. (2.24)℄

forthe q polynomialsinthenon-uniform lattie x(s)=

1 q

s

.

3.3.2 The hypergeometri representation

We start with the ; Jaobi/Jaobi family, i.e., the ase when = ba(x a

1 )(x a 2 ) and ?

=ba ? (x a ? 1 )(x a ? 2 )ba

? a 1 a 2 b a ? a ? 1 a ? 2

6=0. Theotherases anbeobtainedinasimilarway. Then,

substitutinginthe above expressionwe ndthattheq lassialpolynomialsbeomes

P n = r n (baa 1 a 2 ) n (x=a 1 ;q) n (x=a 2 ;q) n (1 q) n x n 3 ' 2 q n ;a ? 1 x 1 ;a ? 2 x 1 q 1 n a 1 x 1 ;q 1 n a 2 x 1 q; b a ? b a q n+3 ! :

From the last formula it is not easy to see that P

n

are polynomials on x of degree exatly equal

n, thus, we will applyto the above equation the transformations (3.2.5) and (3.2.3) given in[10 ,

page 61℄. Notie that we an apply the transformation formula (3.2.5) [10 , page 61℄ beause

the polynomials and q ?

have the same independent term, and then the ondition baa

1 a 2 = qba ? a ? 1 a ? 2

isfullled. So,thehypergeometrirepresentationofthemoniq lassial; Jaobi/Jaobi

polynomialsis P n (x)= a n 2 (a ? 1 =a 2 ;q) n (a ? 2 =a 2 ;q) n (a ? 1 a ? 2 a 1 1 a 1 2 q n 1 ;q) n 3 ' 2 q n ;a ? 1 a ? 2 a 1 1 a 1 2 q n 1 ;x=a 2 a ? 1 =a 2 ;a ? 2 =a 2 q;q ! : (3.7)

Notiethat,sineand ?

areinvariantwithrespetto thehangea

1 ()a

2 anda

?

1 ()a

?

2 ,then

wean obtainan equivalent hypergeometrirepresentation

(17)

NotiealsothatfromanyoftheabovetwoformulasfollwosthatP

n

isapolynomialofdegreeexatly

equaln. Beforestart withthedetailedstudyof eah ase letuswriteanotherequivalent form for

the; Jaobi/Jaobipolynomialswhihan beobtainedapplyingthetransformation(III.12)from

[10 , page241-242℄ to (3.7):

P

n (x)=

q

n

2

( a ?

2 )

n

(a ?

1 =a

2 ;q)

n (a

?

1 =a

1 ;q)

n

(a ?

1 a

?

2 a

1

1 a

1

2 q

n 1

;q)

n

3 '

2 q

n

;a ?

1 a

?

2 a

1

1 a

1

2 q

n 1

;a ?

1 =x

a ?

1 =a

2 ;a

?

1 =a

1

q;qx=a ?

2 !

: (3.9)

If we nowhoose =aq(x 1)(bx ) and ?

=q 2

(x aq)(x q), then Theorem2.1 and

Eq. (3.7) gives,fortheweight funtionand the polynomials,respetively

!(x)=

(x=a;x=;q)

1

(bx=;x;q)

1

; p

n

(x;a;b;;q)= (aq;q)

n (q;q)

n

(abq n+1

;q)

n 3

'

2 q

n

;abq n+1

;x

aq;q

q;q

!

;

i.e.,theBigq Jaobipolynomials. Ifwenowhoose=q N 1

theybeomestheq Hahn

polyno-mialsQ

n

(x;a;b;Njq) (usuallytheyarewrittenaspolynomialsinx=q s

,see[13 ,18 ℄). Obviously,

ifwe useinstead of formula(3.7) theformulas (3.8) and (3.9) we obtainother representations for

theBig q Jaobipolynomials.

Fortheother11asesweandothesame,substitutethepolynomialsand ?

in(3.6)andmake

theorresponding alulations, buthere we willshow how, from theq lassial; Jaobi/Jaobi

polynomials,an be derived all other ases bytaking the appropiatelimits. A similarstudyhave

beendonein[23 ℄. Herewewillompleteit. Wewillgivethedetailsonlyinsomespeial\diÆult"

ases orwhen thelarityand theauray arerequired.

We ontinue with the q lassial; Jaobi/Laguerre polynomials. To obtainthem we take

thelimita ?

2

! 1. Then,=ba(x a

1

)(x a

2 ) and

q ?

=qba ?

(x a ?

1 )(x a

?

2 )=qba

?

a ?

2 (x a

?

1 )(x=a

?

2

1)= baa

1 a

2

a ?

1

(x a ?

1 )(x=a

?

2

1)! b aa

1 a

2

a ?

1

(x a ?

1 );

wheretherelationbaa

1 a

2 =qba

?

a ?

1 a

?

2

hasbeenused. Inthisase and sine

lim

a ?

2 !1

(a ?

1 a

?

2 a

1

1 a

1

2 q

n 1

;q)

k

(a ?

2 =a

2 ;q)

k

=q (n 1)k

a ?

1

a

1

k

;

Eq. (3.7) beomes

P

n (x)=

a

1 a

2

a ?

1

n

(a ?

1 =a

2 ;q)

n q

n(n 1)

2 '

1 q

n

;x=a

2

a ?

1 =a

2

q; q

n

a ?

1 =a

1 !

: (3.10)

If we hoose now = (x 1)(x+b) and ?

= q 2

(x bq), then we obtain the q Meixner

polynomials

M

n

(x;b;;q)=( ) n

(bq;q)

n q

n 2

2 '

1 q

n

;x

bq

q;

q n+1

!

:

Inthisase !(x)=

(x=b;q)

1

( x=b;x;q)

1

. Puttingintheabove formulas b=q N 1

and= p 1

we arrive

to theQuantumq Kravhuk polynomials K qtm

n

(x;p;N;q).

The next family is the q lassial ; Jaobi/Hermite one. In this ase we take the limit

a ?

1 ;a

?

2

!1. Then, =ba(x a

1

)(x a

2

) and q ?

=ba ?

a

1 a

2

,thus(3.7) beomes

P

n

(x)=( a

2 )

n

q

n

2

2 '

0 q

n

;x=a

2

|

q;q

n

a

2 =a

1 !

(18)

Choosing=(x a)(x 1) and q ?

=aweobtaintheAl-Salam &Carlitz IIpolynomials

V (a)

n

(x;q)=( a) n

q

n

2

2 '

0 q

n

;x

0

q;

q n

a !

;

Ifnow=(x i)(x+i)andq ?

=1,we arrivetotheDisreteq HermitepolynomialsII e

h

n (x;q)

e

h

n

(x;q)=i n

2 '

0 q

n

;ix

|

q; q

n !

=x n

2 '

1 q

n

;q n+1

0

q

2

; q

2

x 2

!

;

and fortheweight funtionwehave !(x)=(ix; ix;q) 1

1

=( x 2

;q 2

)

1 =

Q

1

k=0 (1+x

2

q 2k

)

1

.

Theq lassial; Laguerre/Jaobipolynomials. Inthisasea

2

!1. Then,= qba ?

a ?

1 a

?

2 a

1

1 (x

a

1

) and ?

=ba ?

(x a ?

1

)(x a ?

2

), thusEq. (3.9) gives

P

n

(x)= ( a ?

2 )

n

q

n

2

(a ?

1 =a

1 ;q)

n2 '

1 q

n

;a ?

1 =x

a ?

1 =a

1

q;qx=a ?

2 !

= a n

1 (a

?

1 =a

1 ;q)

n (a

?

2 =a

1 ;q)

n3 '

2 q

n

;x=a

1 ;0

a ?

1 =a

1 ;a

?

2 =a

1

q;q

!

:

(3.12)

The last equality follows from the Jakson transformation formula (see [10 , Eq. (III.5), page

241℄), or, diretly, taking the limit in formula (3.8). If we now hoose = aq(x 1) and

?

=q 2

(x aq)(x q),we obtainthe Bigq Laguerre polynomials

p

n

(x;a;;q)= (aq;q)

n (q;q)

n3 '

2 q

n

;0;x

aq;q

q;q

!

= (aq;q)

n ( q)

n

q

n

2

2 '

1 q

n

;aqx 1

aq

q;

x

!

:

Notie that they are nothing else that the Big q Jaobi when b = 0. Here !(x) =

(x=a;x=;q)

1

(x;q)1 .

To this lass also belong the K aff

n

(x;p;N;q). In fatthey are Big q Laguerre polynomials with

parameters a=q N 1

and =p.

The q lassial ; Hermite/Jaobi polynomials. In this ase a

1 ;a

2

! 1, thus = qbaa ?

1 a

?

2

and ?

=ba ?

(x a ?

1

)(x a ?

2

). Then,from Eq. (3.9) one easilynd

P

n

(x)=q

n

2

( a ?

2 )

n

2 '

1 q

n

;a ?

1 =x

0

q;qx=a ?

2 !

: (3.13)

Nowhoosing=aand q ?

=(x 1)(x a), (3.13) leadstotheAl-Salam&CarlitzIpolynomials

U (a)

n

(x;q) =( a) n

q

n

2

2 '

1 q

n

;x 1

0

q;

xq

a !

:

Inthisasetheq weightfuntiontakestheform!(x)=(qx=a;qx;q)

1

. Ifweputa= 1,thethe

Al-Salam &CarlitzI polynomials beomesthedisreteq HermitepolynomialsIh

n (x;q).

(19)

To obtain the 0 Bessel/Jaobi polynomials we will take the limit a 1 ;a 2 ;a ? 2

! 0. Thus,

= bax 2

and ?

= ba ?

(x a ?

1

)x, but now we have a problem taking the limit in the expression

(a ? 1 a ? 2 a 1 1 a 1 2 q n 1 ;q) k

,sowewillobligedtheparametersa

1 ;a 2 ;a ? 2

tendtozerosuhthata ? 2 a 1 1 a 1 2 = q Æ

,withÆ axedonstant suhthatq Æ

=ba=(qba ?

a ?

1

). Then,takingthelimitinEq. (3.7) weobtain

P n (x)= q n 2 ( a ? 1 ) n (q n+Æ 1 ;q) n 2 ' 1 q n ;q n+Æ 1 0 q;qx=a ? 1 ! ; q Æ = b a qba ? a ? 1 : (3.14)

To thislassbelongstheAlternative q CharlierpolynomialsK

n

(x;a;q). Infat, putting=ax 2

and ?

=q 2

x(1 x), thusq Æ

= aq and then

K

n

(x;a;q)= ( 1) n q n 2 ( aq n ;q) n 2 ' 1 q n ; aq n 0 q;qx ! :

Forthem we have !(x)=jxj (x 1 ;q) 1 1

,where q

= a=q.

For the0 Bessel/Laguerrepolynomialswe have thelimita

1 ;a 2 ;a ? 1

!0and a ?

2

!1. Thus,

=bax 2

and ?

=ba ? (x a ? 1 )(x a ? 2 )=ba

? a ? 2 (x=a ? 2 1)(x a ? 1 ) =baa

1 a 2 a ? 1 1 q 1 (x=a ? 2 1)(x a ? 1 ).

Ifwenowtake thelimitinsuh away thata ? 1 a 1 a 2 = q Æ

we arriveto thefuntion ?

=baq Æ 1

x.

In thisase Eq. (3.9) immediatelygives

P

n

(x)=q

n(n+Æ 1) ( 1) n 1 ' 1 q n 0 q; q n+Æ x ! ; q Æ = b a a ? q : (3.15)

Now, setting =x 2

and ?

=q 2

x,we have q Æ

= q and we obtaintheStieltjes-Wigert

polyno-mials

S

n

(x;q)=( 1) n q n 2 1 ' 1 q n 0 q; xq n+1 ! :

Here !(x)= p x log q x 1 .

The0 Jaobi/Jaobipolynomials. Inthisasethelimitisa

2 ;a

?

2

!0providingthata ? 2 =a 2 = q Æ

,then=bax(x a

1 ),

?

=ba ?

x(x a ?

1

)and (3.7) gives

P n (x)= q n 2 ( a ? 1 ) n (q Æ ;q) n (a ? 1 =a 1 q Æ+n 1 ;q) n 2 ' 1 q n ;a ? 1 =a 1 q n+Æ 1 q Æ q;qx=a ? 1 ! ; q Æ = b a a 1 qba ? a ? 1 : (3.16)

Putting=ax(bqx 1) and ?

=q 2

x(x 1),q Æ

=aq,thus

p

n

(x;a;bjq)= ( 1) n q n 2 (aq;q) n (abq n+1 ;q) n 2 ' 1 q n ;abq n+1 aq q;qx ! ;

whih are nothing else that the Little q Jaobi polynomials. If now we take = px(1 x),

? =q 2 x(x q N

) wearrive to thefollowingexpression

K

n

(x;p;N;q)= ( 1) n q nN+ n 2 ( pq N+1 ;q) n ( pq n ;q) n 2 ' 1 q n ; pq n pq N+1 q;xq N+1 ! ;

thats onstitutes an alternative denitionfor the q Kravhuk polynomials whih is equivalent to

the\more"standard one justusingthetransformation formula (III.7)from [10 , page241℄

K

n

(x;p;N;q) = (q N ;q) n ( pq n ;q) 3 ' 2 q n

(20)

Finally, we have !(x)= jxj (qx;q)1 (qbx;q)1 ,q

=a and !(x)= jxj (q N+1 x;q)1 (x;q)1 , q =pq N

for theweight

funtions of theLittleq Jaobiand q Kravhukpolynomials,respetively.

The0 Jaobi/Laguerrepolynomials. Inthisasewetakethelimitisa

2 ;a

?

2

!0anda ?

1 !1

insuha waythata ? 2 =a 2 = q Æ

,so =bax(x a

1 ),

?

=baa

1 q

Æ 1

x=a ?

x,and then

P

n

(x)=( a

1 ) n q n(n+Æ 1) 2 ' 1 q n ;x=a 1 0 q; q n+Æ ! ; q Æ = baa 1 qa ? : (3.17)

Putting=ax(x+1) and ?

=q 2

x,thenq Æ

= aq,andweobtaintheq Laguerrepolynomials

L

n

(x;q)L

n

(x;a;q)

L

n

(x;a;q)=( 1) n q n 2 a n 2 ' 1 q n ; x 0 q;aq n+1 !

; !(x)= jxj

( x;q)

1 ; q

= a:

If we now hoose = x(x 1) and ?

= q 2

ax, we obtain q Æ

= q=a and then we arrive to the

q Charlierpolynomials

C

n

(x;a;q)=( 1) n q n 2 a n 2 ' 1 q n ;x 0 q; q n+1 a !

; !(x)= jxj (x;q) 1 ; q =a 1 :

The0 Jaobi/Besselpolynomials. Herewetake thelimitisa

2 ;a ? 1 ;a ? 2

!0insuhawaythat

a ? 1 a ? 2 =a 2 =q Æ

,so=bax(x a

1 ),

?

=ba ?

x 2

=baa

1 q

Æ 1

x 2

,and then(3.7) gives

P

n

(x)=q

n(n+Æ 1) (q n+Æ 1 =a 1 ;q) 1 n 2 ' 0 q n ;q n+Æ 1 =a 1 | q;xq 1 Æ ! ; q Æ = b aa 1 qba ? : (3.18)

This family does not appearin theq Askey Sheme unless they are nota trivial limitof a more

general q family. We will take the following parameterization = ax(x b) and ? = q 2 x 2 . Then, q Æ

=abq and we obtainthatthis0 Jaobi/Bessel polynomials,denotedbyj

n

(x;a;b)

j

n

(x;a;b)=(ab) n q n 2 (aq n ;q) 1 n 2 ' 0 q n ;aq n | q;x=(ab) !

; !(x)=jxj (bq=x;q) 1 ; q =a 1 q 5 :

They maindata areshown inTable 3.3.2.

The 0 Laguerre/Jaobi polynomials. In this ase a

2 ;a

?

2

! 0, a

1

! 1, q Æ = a ? 2 =a 2 , then

=a x=ba ? a ? 1 q Æ+1 x, ?

=ba ? x(x a ? 1 ),and P n

(x)=( a ? 1 ) n q n 2 ( q Æ ;q) n2 ' 1 q n ;0 q Æ q;xq=a ? 1 ! ; q Æ = a ba ? a ? 1 q : (3.19)

Putting = ax and ?

=q 2

x(x 1), thus q Æ

= aq, and we obtainthe Little q Laguerre or

Wallpolynomials

p

n

(x;ajq)=( 1) n q n 2 (aq;q) n2 ' 1 q n ;0 aq q;qx !

; !(x)=jxj (qx;q) 1 ; q = a:

Finally,the 0 Laguerre/Bessel familyfollows from Eq. (3.8) taking thelimit a

1 ;a ? 1 ;a ? 2 ! 0 and a 2

!1 providing thata ? 1 a ? 2 =a 1 = q Æ

,thus=ax=ba ?

q Æ+1

x, ?

=ba ? x 2 and P n

Figure

Figure 1. Re
urrent s
hema using the q dilation.

Figure 1.

Re urrent s hema using the q dilation. p.6
Table 3.3.2: The q 
lassi
al polynomials j

Table 3.3.2:

The q lassi al polynomials j p.21

Referencias

Actualización...