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
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
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
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 ?
( ?
() 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)
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
?
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)
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 ?
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
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
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
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
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
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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;
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
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
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
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 !
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).
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
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
