ETNA
Electronic Transactions on Numerical Analysis.
Volume 27, pp. 34-50, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
FACTORIZATION OF THE HYPERGEOMETRIC-TYPE
DIFFERENCE EQUATION ON THE UNIFORM LATTICE
R. ALVAREZ-NODARSE , N. M. ATAKISHIYEV , AND R. S. COSTAS-SANTOS Abstract. We discuss factorization of the hypergeometric-type difference equations on the uniform lattices and
show how one can construct a dynamical algebra, which corresponds to each of these equations. Some examples
are exhibited, in particular, we show that several models of discrete harmonic oscillators, previously considered in a
number of publications, can be treated in a unified form.
Key words. discrete polynomials, factorization method, discrete oscillators
AMS subject classifications. 33C45, 33C90, 39A13
1. Introduction. The study of discrete system has attracted the attention of many authors in the last years. Of special interest are the discrete analogs of the quantum harmonic
oscillators [2, 5, 6, 9, 11, 12, 16, 17, 18, 21, 25] among others.
There are several methods for studying such systems. One of them is the factorization
method (FM), first introduced for solving differential equations [28, 19]. This classical FM is
based on the existence of the so-called raising and lowering operators for the corresponding
equation, which allow to find the explicit solutions in a simple way, see e.g. [7, 22]. Later
on, Miller extended it to difference equations [23] and -differences –in the Hahn sense–
[24]. In the case of difference equations this method has been also extensively used during
the last years (see e.g. [9, 11, 14, 22, 29] for difference analogs on the uniform lattice and
[4, 5, 6, 9, 11, 12, 15] for the -case).
Later on, references [7, 8, 13] indicated a way of constructing the so-called “dynamical
symmetry algebra” by applying the FM to differential or difference equations [3, 11, 12] and
then this technique has been used to consider some particular instances of -hypergeometric
difference equations. Of special interest is also the paper by Smirnov [29], in which the equivalence of the FM and the Nikiforov et al formulation of theory of -orthogonal polynomials
[26], was established. In [4], following the papers [15, 22] for the classical case, it has been
shown that one can factorize the hypergeometric-type difference equation (2.1) in terms of
the above-mentioned raising and lowering operators.
Our main purpose here is to show how to deal with all different cases of difference
equations on the uniform lattice in an unified form. One should consider this paper
as an attempt to provide a background for the more general -linear case (since in the limit
as goes to , the -linear case reduces to the uniform one). Some results concerning this
general case will be also given in the last section.
The structure of the paper is as follows. In Section 2 some necessary results on classical
polynomials are collected. In section 3 the factorization of the hypergeometric-type difference equation is discussed, which is used in section 4 to construct a dynamical symmetry
algebra in the case of the Charlier polynomials. In section 5 the Kravchuk and the Meixner
cases are considered in detail. Finally, in section 6 we briefly discuss a possibility of applying
this technique to the -case.
Received June 16, 2003. Accepted for publication April 22, 2004. Recommended by F. Marcellán.
Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain
(ran@us.es).
Instituto de Matemáticas, UNAM, Apartado Postal 273-3, C.P. 62210 Cuernavaca, Morelos, México
(natig@matcuer.unam.mx).
Departamento de Matemáticas, E.P.S., Universidad Carlos III de Madrid, Ave. Universidad 30, E-28911,
Leganés, Madrid, Spain (rcostas@math.uc3m.es).
34
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35
FACTORIZATION OF DIFFERENCE EQUATIONS
2. Preliminaries: the classical “discrete” polynomials. The discretization of the hypergeometric differential equation on the lattice [26, 27] leads to the second order difference equation of the hypergeometric type
%
&
"(' "$#
%
&
)+*-,
!
/.
.
.
.
.
.
102 " !
, 304 5(
.
where
The most simple lattice is the uniform one + and it corresponds to the equation
"6# !
"(' 7*-8
(2.2)
(2.1)
The above equation have polynomial solutions 9:;
, usually called classical discrete or'
' :<=?>@ #-AB" >C!
D A A EF
.
thogonal polynomials, if and only if It is well known [26] that under certain conditions the polynomial solutions of (2.2)
are orthogonal. For example, if DGH!
&JI;K
S* , for all TUS*-,V,WX,Y8V8Y8 , then the
polynomial solutions 9:Z
of (2.2) satisfy
[%9 : ,&9\^]&_
(2.3)
bR `Za
LNMPO
KK LNMPOQ R
9 : X9\/
cGd
7e : \^fF:g ,
where the weight functions Gd
are solutions of the Pearson-type equation
h
" !
DGH " !
@ !
"6# kjlGH!
m8
: "on : :
In the following we will consider the monic polynomials, i.e., 9 : !
)7
(2.4)
ih
!
DGH
kjP # DGH!
or
`;a
"qpYpVp .
The polynomial solutions of (2.2) are the classical discrete orthogonal polynomials of
Hahn, Meixner, Kravchuk and Charlier and their principal data are given in Table 2.1.
TABLE 2.1
The classical discrete orthogonal monic polynomials.
rds-t%umv
wFxcs y z t%u!{D|/v
}6s ~y  t%uYv
Meixner
Kravchuk
Charlier
ƒ „X…N† ‡
ƒ ˆE… | ‡
ƒ ˆ…&‰ v
ƒ ˆE… |+Š6‹ ‡
ƒ ˆ…&‰ v
Πt%umv
ut|ŠŽ<Žumv
u
u
u
 t%uYv
tl‘’Š“‹mv”t|U•‹mvZ–t—˜ŠC‘Š•™!vNu
tlš˜›‹mvNuŠœš-
š/œu
Œ Š 
t%uŠŽ‘’Š“‹mv”t|¢›‹Žumv
šduŠœBš
¥ s
¦ t ¦ Šœ/ŠC‘Š§‹mv
tk‹¨šHv ¦
© t%umv
Eª « ž¬ xcŸc N­ ªJ« z ¬ ¡ ¬ ¡ ­
ªE« ž ŸX N­ ªE« ¬ ­
µH¶·¸¹Bº»¶¼¾½£¿/¹1º
sEË Jª « x ¬ z ¬Ìž¬ s ¬ ¡ ­ x ¬ z ¬ Ê s ¬ Ê Ÿ ¡
« ž Ÿ s Ÿ ¡ ­ Ë « x ¬ z ¬ s ¬ ¡ ­ÈÍ1¯ x ¬ s ¬ ¡ °
!® ªE« ~ ¬ N­
Jª « ~Y­ ªE« ¬ ¡ ­
ÀÁ<Â!¶Ã’Ä’ÅlÂ!¶ º&Æ
sË « ~Y­ÈÍE Í
« ¡ ŸXV­ÏÎmÐÑ Í
ž ¡ !Ÿc ŸF
 ¡  t%u@£|¤v
ŸF
s
¡ ŸF
ž tk‹@ v ž ŸX ¯ °N±
±
Ç^ÄÅÈÂ!¶&º&Æk¶¼½5¿/¹1º
ž s tk‹@ v s
¯s °±
±
ÉsÊ
Hahn
€s t%umv
‚1s  t%umv
š
¦
²&³´  ®
ªE« ¬ ¡ ­
þÁ<Â
¦;Ò š s
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R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
They can be expressed in terms of the generalized hypergeometric function Ó F Ô ,
shifted factorial)
b
Ó F PÔ Õ+Ö n a , , Ö n g V, ,Y8Y8V8Y8Y8V8Y,, nÖ Ô Ó KK Ø×iÚÙ
KK
a g
I MZÛ
Ö I Ö g I Vp pYp Ö Ó I HI
a
,
n I n g I pYYp p n ÔY
I dT Ü
a
Ö ¢
F,Ý Ö I Ö Ö " ! Y Ö " F
pYpVp Ö " T’!
m,5T/=,WX,”Þ-,Y8V8Y8•8
Û
Using the above notations, we have for the monic polynomials of Hahn, Meixner, Kravchuk
and Charlier, respectively
ßÌà
D ?“â›
:Zã " ! D:
æF,&ä " ã " > " " ,V?> K × ,
: Q á F,&â›
@ ä " ã " > " :6å F g Õ
KK
?§œ
â ,&ã
K
çè
êZ
:cë :
?>,Yæ K ? ×¾,
: Q é @ ë ì
: g F a Õ
KK
ê
ë
K
í Ó
DîH
: â6Ü
?>,Væ K ×¾,
: @ %âï§>P
»Ü g F a Õ 5â
KK î
K
ð
:
: é !
)UD ë >,Yæ K × 8
?
KK ë
g FÛ Õ
K
A further information on orthogonal polynomials on the uniform lattice can be found in
[1, 20, 26, 27].
3. Factorization of the difference equation. Let us consider the following second order linear difference operator
h
ñ !
@=æòZ5!
có `dô ® 6òZ!
-ó ô ® " E "$# NjlõÌ,
a
. "
h
à .
where ó ôL ö
ä÷
for all ä¢øù , òZ
¾ûú " "$# kj , and õ
identity operator, and let ü1:-
: be the set of functions
(3.1)
ü1:Z
@
(3.2)
is the
ú dG 9:;
m,
fF:
where fB: is a norm of the polynomials 9÷:Z
, which satisfy equation (2.2), and Gd
is the solution of the Pearson-type equation (2.4). If 9:;
possess the discrete orthogonality property
(2.3), then the functions ü1:Z!
have the property
[ü3:Z
m,Wü \ ] _ Using the identity
(3.3)
i.e., the functions
we will refer to ñ
a
bR `;a
LNMPO
ü3:Z
&ü \ 7eY: \ 8
Q
and the equation (2.2), one finds that
ñ &ü3:;
@ ' :Øü3:;
m,
a
ü1:Z
, defined in (3.2), are the eigenfunctions of ñ .
a
as the hamiltonian.
In the following
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FACTORIZATION OF DIFFERENCE EQUATIONS
37
n
first step is to find two operators and !
such that the Hamiltonian ñ !
^
Ö
n Our
n
a
, i.e., the operators Ö and factorize the Hamiltonian ñ .
Ö DEFINITION
a
3.1. Let ä be a real number. We define a family of ä -down and ä -up
operators by
ýXþ
(3.4)
ý
à
à c02ió ` ô ®5ÿ ó ô ® ú !
ú "$# -õ˜,
à
à c02 ÿ ú !
&ó `dô ® ú "6# -õ ó ô ® ,
respectively.
A straightforward calculation (by using the simple identity
all ä6ø
ñ @
a
ó ô® ) shows that for
à à »,
ý
ýXþ
i.e., the operators à and à factorize the Hamiltonian, defined in (3.1). Thus, we have
the following
ý þ
ý
à and
T HEOREM 3.2. Given a Hamiltonian ñ a , defined by (3.1), the operators
à !
, defined in (3.4), are such that for all ä$øŽù , the relation ñ !
) à à ýþ holds.
a
ý
ý þ a dynami4. The dynamical algebra: The Charlier case. Our next step isý to find
cal symmetry algebra, associated with the operator ñ , or, equivalently, with the correa
and n , thath factorize
sponding family of polynomials, i.e., To find two operators
n
Ö
n the
hamiltonian ñ , i.e., ñ !
ö
, and are such that its commutator », kj)
Ö the identity operator.
Ö
n ÷ n a @+õ , awhere õ denotes
Ö THEOREM
Ö 4.1. Let ñ be the hamiltonian,
n
defined in (3.1). The operators ’
a
à !
and !
à ,h given in (3.4), factorize the Hamiltonian ñ (3.1) and satisfy
Ö
», n NjZ for a certain complex numbera , if and only if the
the
ý commutation relation
ý þ
Ö
following two conditions hold:
h
5“äh §ä÷
"$# æ§ä÷
kj
U
æ!
"$# !
Nj
and
æ§ä " " 5“ä
"6# 5“ä÷
÷“
÷ # æ8
à and à , a straightforward calcu"
Ö
`Zô ® å þ !
õ , where
g
Ö
Ö
a
þ
" ?“ä
h §ä " "6# 5“ä " !
m,
!
)¢ ú a
h
"$# æ“ä
Y,
!
)¢ ú æ§ä÷
“ä
g
" ?§ä÷
" æ§ä÷
"$# æ§ä÷
»8
!
) å
In the same way, à à @=ñ ó ô ® " g ó `Zô ® " Dõ , where
Ö
Ö
a
å
a
þ
"6# !
m8
@=æòZ
m,
=æòZ5(
»,
+E!
g
a
å
Proof. Taking the expression
for the operators
" Ö ó
lation shows that à !
à !
) ó ô ®
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R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
Consequently,
h
à m, à !
NjZ ÿ F
ó ô ® " ÿ !
F
ó `Zô ® " ÿ õÌ8
g
g
Ö
Ö
a
a
å
å
þ
To eliminate the two terms in the right-hand side of (4.1), which are proportional to P ,
æ * and g g æ * . But ” æL one have to require that !
"
"
a
a
a
a
&
g !
, hence, the requirement that @
!
entailsh the
relation g g
a
a
, and vice versa. Thus, from (4.1) it follows that the commutator à », à kj@
,
g
Ö
Ö
iff @
!
and ÷
!
@
.
þ
a
a
å
å
Using the main data for the discrete polynomials (see Table 2.1), we see that the only
"
#
!
^
>P »8 and
possible solution of the problem 1 corresponds to the case when !
ä–+* , i.e., the Charlier polynomials. Moreover, in this case ' : > .
(4.1)
C OROLLARY 4.2. For the hamiltonian, associated with the Charlier polynomials,
ñ! !
)¢#" ë ó `Zô ® ú " ë ó ô ® " " ë DõÌ,
a
ñ
a
!
”ü : 7>ü : m,
ü : $
ó `Hé ë L”` : ð
: é »,
FÜ&>Ü
ë&% *Ø,
>•+*-,YF,”-,Y8Y8V8»8
Furthermore, the operators
" " ;ó ô ® " ë õØ,
@ " !ó `dô ® " ë õÌ,
Û
Û
ý þ ý
h
are such that ñ' !
and Ö », Ö NjZ¢ .
' Û
Û
Û
Û
Notice that,a since
ý ñ a ý þ
&ü^
ü^þ ,
ñ !
)( &ü^+
*^
!
+
( !
ӟ^
+*^ ÷(
+( ӟ^
+*
Û
Û
Û
Û
Û
Û
Û
a
ý þ
ý þ ý
ý þ
ý ' !ý )
þ ( ý &þ ü^+
*F,
Û ñ !)
( &ü^+
*^
!
ý þ &ü^
Y '" !
ӟ^
Û
Û
Û Û
Û
a
ý
ý '" !ý )
þ ( ý &ü^+
*F8 ý
Û
ý
In other words, if ü^
is an eigenvector of the hamiltonian ñ !
, then &ü^
is the
'
Û
a
eigenvector of ñ !
, associated with the eigenvalue $ , and &ü^!
isý þ the eigenvector
h
'P" . In general then j—Û I-
&ü^!
and h j—I-
&ü^!
a
of ñ , associated with the eigenvalue
ý
Û
a eigenvectors corresponding to the eigenvalues ' 6T andÛþ '" T , respectively.
are also
ý
ý
Using the preceding formulas for the Charlier polynomials, one finds
”ü : , : ü :- m,
&ü : . : ü : »,
(4.2)
ý
Û
a
Û
ý þ
`;a
where ,@: and ./: are some constants.
If we now apply to the first equation of (4.2) and then use the second one and (3.3),
'
Û , : . On the other hand, applying !
to the second equation in
we find that :$//
. ý :!@
Û
`Za
(4.2) and using the first one, as well as the fact that ý &þ ü : !
)U
' : " &ü : !
, one
+
"
'
'
'
Û
Û
obtains that :“0
, :1/
. :- :- , from which
ý þ itý follows that : should be a linear
a
a
function of > (that is also obvious from Table 2.1).
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FACTORIZATION OF DIFFERENCE EQUATIONS
If we use the boundary conditions NGd
tion by parts, we obtain
KK LNMPO!Q R *
&ü3:P
] _ ,
Û
Û
ý þ
ý
i.e., the operators and are mutually adjoint.
Û
Û
From the above
property) and (4.2) it follows that ./:2- ,@: ,
ý þ equalityý (the adjointness
'
a
thus , : g :- , therefore ,@:/ ú ' : - and ./:¤ " ' : , i.e., we have the following
a
a
C OROLLARY 4.3. The operators !
and !
are mutually adjoint with respect to
p p
Û
Û
the inner product [ , ] _ and
ý
ý þ
&ü : @43 " Pó ô ® " ë õ15ü : !
) " > " ü 2: - !
m,
a
Û
ý &ü @ 3 " " ;ó ® " ë õ 5 ü " >¾ü
ô
:
: »8
:
`Za
Û
ý þ
[
&ü \ »,»ü3:;
&] _ U[ü \ !
m,
, as well as the formula of summa-
From the above corollary one can deduce that
"
" ü
Û
" !
" ë ü
Using the orthonormality of ü
Û
Û
*
, one obtains that â
ü
6
Û
Û
@+â
7ó `Hé7 g
ë L
8
Û $ F Ü
. Thus
:; ó H` é ë L
h "
ü : !
Nj : ü @
;ó ô ® " ë õ2:
$
8
" "
F Ü=<
Û
> Ü Û
> Ü98
ý
h
Notice that
h
ñ !
m, kj; " ë ë !
" ë »,
ñ !
m, NjdU " ë ë !
÷ ë »8
Û
Û
Û
Û
a
a
ý
ý
ý þ
ý þ
This example constitute a discrete analog of the quantum harmonic oscillator [9].
5. The dynamical algebra: The Meixner and Kravchuk cases. From the previous
results we see that only the Charlier polynomials (functions) have a closed simple oscillator
algebra. What to do in the other cases? To answer to this question, we can use the following
operators:
"
"6# 5!
Xó ` a ô ® ,
a ®
g
Ö @ ú Xó g ô ú 5!
"
"$# æ!
»8
a ®
a ®
Ö @ ó ` g ô ú qó g ô ú 5ì!
For this operators
ñ Ö Ö - !
"6# A §
AA 8
a
n
nWe will define a new hamiltonian ñ and operators and g
ð
" > ,
n @ ð n - @ ð - »,
ñ !
) g ñ and
Ö
O
OÖ
g
O a
ð
>
where
and are some constants (to be fixed later on). Notice that from (3.3) it follows
ð ' "?that
>
O
the eigenfunctions of ñ are the same functions (3.2), but the eigenvalues are g :
,
g
O
i.e.,
ð
ñ ”ü3:Z
U g ' : "> &ü3:;!
m8
g
O
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R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
A straightforward computation yields
ð
ñ n n - !
" # A “ A A g "?> ,
g
O
(5.1)
and
hn
(5.2)
ð
», n - NjZ g
O @
'
" ð
" ñ
" a Y% å g
g
g " å Y%æ
O!@
ð g
÷ g E!
"6#
g
O
"6# 5
åg
"6# æ
ga "
a å
g g
-ó `Zô ®
" a m%5 å Ö m, Ö kj; @ g
g
"
"
å mæ
" ñ @ ÷ìg "$#
a
"$# æ
gå
"
$# a
g
õ " a g
-ó `Zô ®
Xó ô ®
ð
# A
g ,
O
ga A A
or, equivalently,
h
a -ó ô ®
g AA # A
åg m
8
The right-hand side of (5.2) suggests us to use the following new operators
ð n
ð ð
Xó ` ga ô ® ú " !
@
% " ! ó `dô ® òZ
-
»,
R
R
O
(5.3)
ð
ð ð
- ú " ! -ó ga ô ® n - @
% " ! 6òZ
Xó ô ® »,
R
R O
" !
Y%
"6# !
&
. So,
where, as before, òZ
ú E
h
ð
ð
ñ »,AE
NjZ= g % A A # A BE
"
g
O
h
ð
ñ »A, !
Njd g A A # A C - ÷
g
O
h
ð ð ÿ A "
E!
mD, kjd
g g a Dó `dô ®
g
O R
ð
O R
ð ð
O
òZ
ñ "
8 g
A "
R
" òZ
ó
ð
% A A # A g > 5 õ2:d A " a »,
O
g
h
ð
A
A
A
"
#
>
ag ñ g !
W3 % O g 5PõEjk,
h
A "
ô ® a ÷ Xò gF
÷6òXgB?!
Njlõ 8
g
3
The above expression leads to the following
A A * , then the operators ñ , E
and T HEOREM 5.1. If g
and (5.3), respectively, form a closed algebra such that
h
, defined by (5.1)
A ð g " ð ð A %*B
3 ñ !
# A ð g > 5 ,
R O
g
O
O
# A ð g - ÷ ð ð A %*B
E3&ñ !
# A ð g > 5 ,
R O
g
O
O
g ÷“ A *F
Y ñ ÷ > IH-8
g
R F
G
A "
A
Observe also that with this particular choice a @7 *F
and
g
" - ð ð ñ !
" A *F
"$# m,
E
R O a
ò g “ò g 5!
@7 A *F
YE
"6# "$#-A »8
ñ »,D
kj; #
g
h
ñ »D, - kjZU
g
h
ð
E
»D, kjZ
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41
FACTORIZATION OF DIFFERENCE EQUATIONS
Furthermore, using the boundary conditions NGd
ð ð bR `Za
" ! &ü3:Z
&ü \ ÷
O R
LDMPO
ð ð bR `Za
" ! &ü3:Z
&ü \ ÷
O R
LDMPO
U[ü3:;D, - ü \ ] _ ,
KK LDMPO!Q R 7*
bR `Za
[JPü3:Z,Wü \ ] _ , one finds
ò;æ(
&ü3:Zæ!
&ü \ LNMPO
ð ð bR `;a
òZ
&ü3:P
&ü \ " O R
LNMPO
i.e., the following theorem follows.
T HEOREM 5.2. The operators E
and are mutually adjoint.
Notice also that the operators ñ and ñ !
are selfadjoint operators.
'
' : =a ?>@ #-Am" %>ö
g œ!
D A A E
, the identity A A 7* is equivalent
R EMARK 5.3. Since '
'
#-A
to the statement that : is a linear function of > . In this case : =?> .
A
A
* , i.e., the case of the
In the following we will consider only the case when Meixner, the Kravchuk and the Charlier polynomials.
If we define the operators
ð
@Sñ !
YD # A g `Za
g
O
ð
ð ð
í
@= # A g E
÷
`
R
O
ð ð
í
A
#
g ÷
-?
@=
R
O
í
(5.4)
then
hí
Û
!
m,
Û
where
kj;
íLK
ð
Uæ -# A A *F
g
R
ð
>
#
A
U D
Û
Û
íLK
a
y
ð
A %*B
E3&ñ ÷ # A g > @
5 ,
O
g
O
ð A
ð
%*B
3 ñ ÷ # A g > 5 ,
O
g
O
hí
í
í
», ?
- NjZ
" ,
`
Û Û
a
ðM -# A ð
" #-A and
D
g Y
% A %*B
6
O
O
ð ðON h
g `Za " g -# A g A % *B
# %*B
÷“%*F
#-A j8
O
R O
The case i* corresponds to the Charlier case (see the previous section).
*,
ð If ÛL+
Û
P #-A
we have ð two possibilities: . In the following we will choose g UöJ ,
*
% * and # A g = .
Û
ÛRQ
O
i.e., ð
>
O
In the first case * one can choose and in such a way that i and 7* .
Û %
R
Û
a
Thus
(5.5)
ð
R
g #-A
h
,
A *F
# A " A % *B
kj
íSK
í
>
h
g A *F
# % *F
@§*F
#-A j
8
R
# A
ð
=
Consequently, the operators
and
are such that
Û
hí
hí
íTK
íLK
(5.6)
»,
kjZ
and
»,
Û
`
í
-?
kj;7
This case corresponds to the Lie algebrað Sp -,DU
.
>
In the second case one can choose
and in such a way that R
Thus
(5.7)
ð
R
g #-A
A %*B
h # A " A *F
Nj ,
>
ð
R
í
Û
Û
m8
æ
h
g A %*B
# *F
÷“%*B
#-A j 8
# A
and a
=*
.
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42
R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
í K
í
and
are such that
Consequently, the operators
hí
hí
íLK
íTK Û
(5.8)
»,
Njd
and -?
»,
Û
í
This case corresponds to the Lie algebra so ÞF
.
Notice that since the operator
adjoint in both cases, i.e.
ñ g
NjZ7
`
í
is selfadjoint, the operators
Û
»8
íTK
are mutually
í
í
[ - ü \ ,»ü3:-] _ =
[kü \ ,
ü3:-] _ 8
`
Dynamical symmetry algebra Sp -,DUì
. Let us consider the first case.
5.1.
with the operator
í
g í
í
í
where
!
, -1!
, and Û
`
tion gives
í
Û
g ÷
í
í
!
Û
-?
í
`
We start
»,
are the operators given in (5.4). A straightforward calcula-
í
g > > !
õØ,
í
>
where is given by (5.5), i.e., the g >
# *F
A *F
÷ #-A %*B
,
E A %*F
Y% A %*B
"$# A is the invariant Casimir operator.
Furthermore, if we define the normalized functions
ü3:Z!
@V
we have
(5.9)
í
GH!
9:P
»,
f g:
í
g ӟ3:;
@ > > !
ӟ3:Z
»,
Û
!
ӟ3:;
@ > "?> &ü3:Z!
m8
Now using the commutation relation (5.6), it is easy to show that
h íLK
í
íLK
&ü3:Z
NjZ > "> 7
!
&ü3:P
»8
Û
Consequently, from (5.9) and the above equation we deduce that
í
í
&ü : @X :- ü :- »,
”ü : @X : ü : !
m8
a
a
`
`;a
In order to compute XÌ: , use (5.9) and (5.10); this yields
> >
" > !
m8
!
@ > "> g ì> "?> ÷ZX g: 6
XÌ:/ ú >@>
In this case the functions kü : : define a basis for the irreducible unitary representation
>
. - of the Lie group (algebra) Sp XD, U
.
From the above formula it follows that the functions ü5:Z
can be obtained recursively
í
via the application of the operator -1
, i.e.,
í :
ú Gd
ü : @
8
pYpYp XØ: - &ü Û », ü Û @
X
f
a
Û
where Gd
is the weight function of the corresponding orthogonal polynomial family and f
Û
is the norm of the 9 .
Û
(5.10)
í
a
m,
í
&ü3:;
@XÌ:Øü3: »8
`Za
í` K
Y
Employing the mutual adjointness of the operators
, one obtains
í
í
WXÌ:¤=[ -ü3:Z
»,»ü3:- ] _ [ü3:Z!
m,
ü3: - ] _ XØ:- ,
a
`
a
a
thus
-?
ӟ3:;
@WXØ:Øü3:-
-
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FACTORIZATION OF DIFFERENCE EQUATIONS
5.1.1. Example: The Meixner functions. Let consider the Meixner functions
è
ç è
—ê;
L
ü\:[ @ ë^] L”` : _ 7 g D5 ë 7 g ;- : V
m,
F Ü >ÜÈêZ
: : Q é
>a`ì*-,
ñ a
[
ú
ê !
ó `Zô ® ú ë " m " Zê ó ô ® " k " ë " ;ê D
-õÌ,
ñ @U ë B " ¨
a
è ð
ð
>
thus ñ [ &ü :[ >ü :[ . In this case we have
»a `Hé , , O a
R
g
a
@ é
@ aW`dé
Therefore
and the hamiltonian
.
"
" Ñ ®
n !
)¢ V ? Zê ë ó Ñ ® "
ó
,
cb ô
`
$
? ë
ë bô
?
"
"
n - @U V æ ga ;ê ë ó Ñ ® " V ga ó Ñ ® 8
b ô
` bô
? ë
ë d
?
Consequently,
ê
ê
ñ ñ "
n n - "
(F8
g
? ë a
Moreover,
ê
ñ &ü :[ !
) ÿ > "
g
í
ü :[ !
m,
ê
" ë
" ë
@= ú B? " ê;
ó ® ú " m " Zê ó ® " ÿ "
Û
? ë `Zô
ë ô
?
í
í
and
-
`
í
" ë
õÌ,
? ë
!
)U
ú F5 " Zê ë ú " m " ê;
® " " ë
ó d` ô ® óô
F " êZ
1õÌ,
? ë
ë
?
ë
?
@U
ú " m " Zê ® " " ë
ë ú B? " êZ
ó `Zô ® óô
F " êZ
3õÌ,
? ë
? ë
? ë
Û
ê
&ü :[ !
@ ÿ > "
í
(5.11)
Using the fact that
finds
*’
í
ü :[ m,
í
ê ÿ ê
g !
ӟ :[ @
( ü :[ m,
&ü :[ !
@ ú > " m%> " êZ
”ü :[ - »,
a
í
"
”ü :[ @=ú >@> ê£(
&ü :[ m8
`
`;a
è
ñ ü [ ’ ü [ !
, together with the formulas (5.4) and (5.11), one
g Û
g Û
`
-
&ü\[ ú Bæ " ê;
Øü\[ æ(
“ ë ` Ñb ü\[ »,
Û
Û
Û
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44
R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
therefore the normalized function ü [
Û
is
D? ë è
ë —ê " ü\[ !
) V e
,
V e L e
—ê;
— ê;
" ! Û
and
ü\:[ V
e
e
? ë è í
:af ë L ê " e
e
3
- 8
V
"
>Ü ê P> " !
hg
A similar result have been obtained before in [9].
5.2. Dynamical symmetry algebra so %ÞB
. Let us consider the second case and define
the following operator
í
í
í
í
í
g g "
!
"
- »8
Û
`
Û
í
í
í
where
, - , and
are the operators given in (5.4). Substituting the value of
>
Û
`
, given by (5.7), and doing some straightforward computations yield
# *F
A *F
÷ # A %*B
í
>
g > > !
õØ,
,
E A %*F
Y% A %*B
"$# A í
i.e., the g is the invariant Casimir operator.
Moreover, if we define the normalized functions as
ü : !
@V
we have
í
GH!
9 : »,
f g:
í
g !
ӟ3:Z
> > !
ӟ3:Z
»,
Û
Now using the commutation relation (5.8), we have
h íLK
í
">
Û
&ü3:Z
NjZ >
&ü3:Z
=> "?> &ü3:;
»8
7!
íLK
&ü3:P
»8
Consequently, from (5.9) and the above equation, we conclude that
í
í
”ü : @WX : ü : - m,
&ü : @X : ü : »8
a
Z` a
í K `
S
Using the mutual adjointness of the operators
, one obtains
í
í
WX : =[ - ü : »,»ü : - ]&_ [ü : !
m,
ü :- ] _ X : - ,
a
`
a
a
thus
(5.12)
í
-
-?
&ü3:;
@XÌ:-
a
í
ü3:- »,
a
To compute XÌ: , use (5.9) and (5.12); this leads to
> >
(
@=%> "?> g " > "> " X g:-
. -
In this case the functions kü : :
> of the Lie algebra so %ÞB
.
a
6
`
ӟ3:;
@XØ:Øü3:
XØ:¤
`;a
!
m8
ú ¾>P
Y> " > (
W8
define a basis for the irreducible unitary representation
As in the previous case, from the above formula it follows
that the functions
í
be obtained recursively via the application of the operator -1!
, i.e.,
í :
can
ú Gd
8
pYpYp XØ: - &ü Û », ü Û @
X
f
a
Û
where GH!
is the weight function for the associated orthogonal polynomial family and f
the norm of the 9 !
.
Û
ü : @
ü : Û
is
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45
FACTORIZATION OF DIFFERENCE EQUATIONS
5.2.1. Example: The Kravchuk functions. Let us consider now the Kravchuk functions
>Ül%âï§>P
»Ü í Ó
üji: $î ] L”` : _ 7 g ?£îd
]lk ` : `dL _ 7 g V
F,”â›
»,
Ül%âï6
Ȇ :
and the corresponding hamiltonian
*nm>amâœ,
ñ a
ú îHB%â 6
ú " ! î " !
Y%â 6!
®
â î " 5“mîH
<
"
®
ñ ¢
ó `Zô
õ¾
óô ,
" ?Cî
" ?Cî
?Cî
a
ð
ð
thus ñ i &ü i: !
) >ü i: . In this case
" ?£î , ú î Z` a , > = k , therefore
O
R
g
a
n @= ú î%âï6 " !
Dó a ® " ú ?£îd
m " !
Nó a ® ,
` g ô
g ô
i
n - @U
@
î%âï6 " a D
ó ga ô ® "
D?CîH
m " a D
ó ga ô ® 8
g
g
@
Consequently,
ñ @ D?CîH
Zñ ÷
g
a
â
â
n n - !
8
Moreover,
ñ &üji: !
@ Õ >C
g
í
â
ü i: »,
× j
U¾ú îD?CîH
Bâ 6 " Dó `dô ® ìú î D?CîH
m " !
Y%â 6!
&ó ô ®
" h â$ î¤ ÷6B»î¤!
NjlõÌ,
ga
í
" !
Dó ® " îú " ! Y%â 6!
ó ® ú î Ø?îH
W5§â›
õØ,
-?
U Ø?îH
Yú B%â 6
`Zô
ô
í
@6îú Bâ 6 " Dó `dô ® " DØ?îd
mú " ! Y%âï6
Dó ô ® ú î Ø?îd
W5§â–
DõÌ,
`
Û
and
í
Û
!
”ü i: @ ÕX>£
í
â
זü i: m,
í
g ӟ i: @
âo
â " F
Øü i: »,
&ü i: @=ú > " !
m⠖
>P
”ü :i - »,
a
(5.13)
í
”ü :i @=ú >@%â §> " &ü :i »8
`
`Za
Using the fact that ñ ü i æ k ü i !
, together with the formulas (5.4) and (5.13), we
g Û
g Û
find
*’
í
-
;
î
îdB%⠓
" ! &ü i @ $
J ü i ÷ V
ü i æ!
,
`
?Cî
?C
î
<
Û
Û
Û
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R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
therefore the normalized function ü i
Û
is equal to
>Ül%â §
>P
Ȇ
ü i “îp] L”` 2: _ 7 g ?£îd
+]lk ` : `dL _ 7 g V
,
F Ül%â 6
!
mÜ
Û
and
ü i: V
AL s
â §>P
»ÜlD?CîH
k ` : í
î
:rq â
- Õ =
8
× Õ
×
:
â6Ü î
?£î
6. The -case. To conclude this paper we will discuss here briefly what happens in the case. The preliminary results, related with this case, have been presented during the Bexbach
Conference 2002 [3]. A more detailed exposition of these results is under preparation.
ü : ’!
ú Gd
9 : ut&E
»,
Bf :
where fB: is the norm of the -polynomials 9÷:;2t”E
, GH
is the solution of the Pearson-type
equation
5 a g
h
NGd
kj; # NGd
or
" !
DGH " !
@7Dææ ë DGH!
m,
n
and is an arbitrary continuous function, not vanishing in the interval , of orthogoÖ
nality of 9: . If 9:Z2t”E
possess the discrete orthogonality property (2.3), then the functions
ü3:Z
satisfy
[kü3:;!
m,Wü \ ])
(6.1)
bR `Za
ü3:Z
ӟ \ !
LDMPO
a
+eY: \ 8
Q
g !
Notice that if ’
/ ú
, then the set ü1:-
D: is an orthonormal set. Obviously, in
a
the case of a continuous orthogonality (as for the Askey-Wilson polynomials) one needs to
change the sum in (6.1) by a Riemann integral [10, 26].
Next, we define the -Hamiltonian v Ô of the form
v
(6.2)
where
w
Ô 304 Bw Ô ,
a
ú æØ ë " D!
ú DæÌ ë D " ! ®
ó Z` ô ® óô
!
!
" Õ D æÌ ë " × õØ8
•
!
Ô 1047
As in the previous case, one can easily check that
v
Ô &ü3:Z!
) ' :Ìü3:Z
»8
Now we define the ä operators:
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47
FACTORIZATION OF DIFFERENCE EQUATIONS
and £
are two arbitrary continD EFINITION 6.1. Let ä be a real number and ’
uous non-vanishing functions. We define a family of ä -down and ä -up operators by
ý þ
(6.3)
ý
à ;
Dææ ë ó ` ô® óô®V V
,
!
x< ’
ú a
;
Dææ ë à ® ú a ®
ó
c02 V ó
V
,
`Zô
x< ô
£
a
à c02
à
¨
respectively. The first result in this case is [3]:
T HEOREM 6.2. Given a -Hamiltonian (6.2) vŽÔ
, then
the operators à !
and
à !
, defined in (3.4), are such that for all ä6øŽù , v Ô à à .
ý þ
with the operator
ý Our next step is again to find a dynamical symmetry algebra,
ý associated
ý þ
v Ô , or equivalently, with the corresponding family of -polynomials.
n
D EFINITION 6.3. Let y be a complex number, and let and be two operators.
Ö
n
We define the y -commutator of and as
h
Ö
n
n
n
Ö », Nj{z5 Ö ÷&y Ö »8
n
We want to know whether the following problem: To find two operators
and n and h », n kÖ j|z˜
and a constant y such that the Hamiltonian v Ô ^
õ , has a
Ö
Ö
non-trivial solution.
n
Obviously, we already know the answer to the first part: these are the operators æ
à !
and !
à , given in (6.3). The answer to the second part of this problem is
Ö in the following two theorems (in what follows we assume that ’
¨!
).
summarized
ý
ý þ
T HEOREM 6.4. [3]Let kü3:Ø
: be the eigenfunctions of v Ô , corresponding to the
'
eigenvalues : : , and suppose that the problem 1 has a solution for S* . Then the
'
P
eigenvalues
the difference
(3.3) are -linear or `;a -linear functions of > , i.e.,
' : ð : " : ð of or
' : ð equation
: " ð , respectively.
g `
a
å [3] Let œ
å -Hamiltonian (6.2). The operators n ) à v ÔE
be the
T HEOREM 6.5.
à !
, given in (6.3) withh £
5}
, factorize the Hamiltonian Ž
v Ô
ý (6.2)
and ?
n kj~ 5
Ö
!
m
,
z
and satisfy the
commutation
relation
for
a
certain
complex
number
y , if and
ý þ
Ö
only if the following two conditions hold
æ!
!
æ§ä÷
D ææ ë " ä÷
a
a
V V
y,
æ§ä÷
æ“ä÷
æ“ä÷
DDææ ë " ! a
̾ä " ! " ÌÌ ë " ä÷
" ÌÌ ë Õ × &y Õ × æ8
̾ä÷
ؾä " ̾
ä
a
a
a
The proof of the theorem 6.5 is similar to the proof of the theorem 4.1, presented here for the
case of the uniform lattice 7 .
Let us point out that the (respectively, `;a )-linearity of the eigenvalues is a necessary
condition in order to provide that the solution of problem 1 exists. But this condition is not
à
sufficient. For example, if we take the discrete -Laguerre polynomials  : €&t E
(for more
'
details see [3]) with U `ZaD7 g , the problem has not a solution, but : is a -linear function
Ö
P
of > .
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R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
6.1. Examples. We present here only two examples, others can be found in [3].
_
6.1.1. The Al-Salam & Carlitz I -polynomials , : ] O pt”E
. We start with the very
well known case: the Al-Salam & Carlitz I polynomials [20]. The corresponding normalized
functions (3.2) are
Í
: Ñ
; ` :
P
,
€
&
t
E
D
Ö `;a
Ö
ü3:Z%H
V
g g‚ a
D?“E
m%1t”E
: cÙ , Ö ,& Ö t&E
Ù
< ú dTFÔ , where TBԜ ga `
Putting ¨Sú
a
ü : : satisfy the orthogonality
condition
ƒ
O
.
*
,& `Za
g
a
1t
Ö
<
,
•047 L 8
, we have that the functions
a ü3:ZH
ӟ \ H
fBÔY +eY: \ ,
Q
T Ô
where the integral „ a d
Df Ô denotes the classical Jackson -integral.
O
For these polynomials
" # !
< Ö , therefore äqï* and y›ï
a
-Hamiltonian has the form
v
Ô @
"
ý þ
ý
:
!
ˆ‡
!
ˆ‡
. The
g ú Ö % <(
m%< Ö ú Ö D?“H
m Ö §d
®
óô
ó `Zô ® g
%^!
g g
"
; " 1%1¨(
c "
Ö §d
&
õØ, £+ L 8
<
%^(
g g
³Í
@+† х a»` Ô Ô _ Ñ ü : and the operators (C L )
] aW`
¡
‰ ÿ ú
%<(JJE
Y< Ö JE
;ó ô ® " Ö õu,
TBÔ»
Û
ý þ
¡
‰ ÿ ú
%<(
m%< Ö ó `dô ® ú Ö J?õ ,
TBÔ»
Û
Consequently, vŽÔE!
ӟ
`Za
ý
are such that
ý
ýØþ
!
)Šv Ô »,
and
h
ý-þ
!
m,
ý
kj Ô ³ b
8
TBÔ
A straightforward calculation shows that the operators and !
are mutually adjoint.
A similar factorization was obtained earlier in [6] and ý more recently
ý þ in [4] with the aid of a
different technique.
ß The special case of the Al-Salam & Carlitz I polynomials are the discrete -Hermite I
:Zp”t E
, C7 L , polynomials, which correspond to the parameter Uö [20].
Ö
6.2. Continuous -Hermite polynomials. Let us now consider the particular case of
the Askey-Wilson polynomials when all their parameters
are equal to zero, i.e., the continuous
ð\‹
-Hermite polynomials [20]. In this case gL.
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FACTORIZATION OF DIFFERENCE EQUATIONS
<ά
!
¤ ú
Let us choose ’
corresponding Hamiltonian is given by
vS
t%umvŽ

.
a
In this case
‚ ™u›
š
?‹
Ÿ1£ Š
›!+v Š œ
t Š#¡ ›!v€¢ ®
ʍ‘ ”’ “–•G—#˜ ’”“–• t — a
’”“–• —
Ê –ž‚Ÿ
p
–ž‚Ÿ
‚š™› Ê Š  ‹
Š
ʍ†’”“–•9— ”’ “–• t — Š Ê¥œ ¦ž‚Ÿ ›!v ’”“–•
˜
ä and y›
E
. The
‚ ™u›
š
£
›!v+ t œ
t d
 ¡ ›!v€¢ ®–¤
’”“–• — p
’”“–• —
Ê –ž‚Ÿ
– ž‚Ÿ
‚š™2›EŸ Ê
›v ¤S¨
t  œ
—
Ê –ž§Ÿ
p
and the ä -operators
ð ‹
a ô®V
ó
g
Ö aC7 g
T Ԛg ©Cª{«d¬ˆ©Dªl«
æ
þ
ð ‹
j
gL
V
Ö Ca 7 g
æT Ôg ©Dªl«\¬9©Dª{« ¬ "
ð ‹
gL
a
®
§
ó
`
ô
œ
g V æ
¬ " €
T Ԁg ©Dªl«d¬ˆ©Dªl«
Ê ­{®2¯ E
ð ‹
j
` gL
ó ` ga ô ® V
œ
æT Ôg ©Dª{«d¬ˆ©Cª{« ¬ ʀ­l®u¯ E
are such that
h
o
` gL
,
¬ ʚœ l­ ®u¯ E
ó aô®,
œ
ʀ­l®u¯ E g
ðj‹
Ö Ö @v Ô and Ö », Ö !
Nj aD7 Ô TBÔ 8
þ
þ
o
Another possible choice is °
¨
[12], hence a straightforward
calculation
ð ‹
T Ô `;a . With this
shows that the two conditions in Theorem 6.5 hold if y i `Za , thus choice the orthogonality of the functions ü?: is „ a ü3:Z
&ü \ fC+eY: \ . In this case, the
Q
`Za
Hamiltonian is
vL
Ÿ £ ®
£ ®
‹ ŠY›
t%umv†Ž ‚  ™ ›
Š
 ² ›Y´ ‹ ›@Š› ³
¢ t Ša
¢
œ ›!v
t  œ › v
Ê
Ÿ ¡ Ž™¶µ ž ’ ™ '— · ¨'¸
4± ’”“¦•9—†’”“–• — Êp •
’”“–• — p
Ê  • ”’ “–•G— ‘
and
"
ðj‹
Õ ó ga
TFÔ ©Dª{«d¬
Xý þ
ð ‹
" 9‡
@
-Õ L ó
T Ô ©Dª{«d¬
D7 g
a
ý
ý
9‡
g
ý þaD7
@
ô ® L “ó ` ga ô ® `dL × ,
` ga ô ® § `ZL ó ga ô ® ד8
With these operators
v
Ô !
@ Ö Ö þ
This case was first considered in [12].
and
h
Ö », Ö !
Nj Ô»³ b þ
o
ðj‹
TBÔ
8
Acknowledgments. We are grateful to A. Ruffing for inviting us to participate at the
Bexbach Conference 2002 and present there our results [3]. This research has been partially
supported by the Ministerio de Ciencias y Tecnologı́a of Spain under the grant BFM-200306335-C03 and the Junta de Andalucı́a under grant FQM-262. The participation of NMA in
this work has been supported in part by the UNAM–DGAPA project IN112300.
ETNA
Kent State University
etna@mcs.kent.edu
50
R. ALVAREZ-NODARSE, N. ATAKISHIYEV, AND R. COSTAS-SANTOS
REFERENCES
[1] R. Á LVAREZ -N ODARSE , Polinomios hipergemétricos y ¹ -polinomios, Monografı́as del Seminario
Matemático “Garcı́a de Galdeano”, Vol. 26, Prensas Universitarias de Zaragoza, Zaragoza, Spain, 2003.
[2] R. Á LVAREZ -N ODARSE , M. ATAKISHIYEVA , AND N. M. ATAKISHIYEV , On a q-extension of the Hermite
polynomials Hn(x) with the continuous orthogonality property on º , Bol. Soc. Mex. Mat., 8 (2002),
pp. 221–232.
[3] R. Á LVAREZ -N ODARSE , N. M. ATAKISHIYEV, AND R. S. C OSTAS -S ANTOS, Factorization of the
hypergeometric-type difference equation on the non-uniform lattices: dynamical algebra, J. Phys. A:
Math. Gen., 38 (2005), pp. 153–174.
[4] R. Á LVAREZ -N ODARSE AND R.S. C OSTAS -S ANTOS , Factorization method for difference equations of
hypergeometric-type on nonuniform lattices, J. Phys. A: Math. Gen., 34 (2001), pp. 5551–5569.
[5] R. A SKEY AND S. K. S USLOV , The q-harmonic oscillator and an analogue of the Charlier polynomials, J.
Phys. A., 26 (1993), pp. L693–L698.
[6] R. A SKEY AND S. K. S USLOV , The q-harmonic oscillator and the Al-Salam and Carlitz polynomials, Lett.
Math. Phys., 29 (1993), pp. 123–132.
[7] N. M. ATAKISHIYEV , Construction of the dynamical symmetry group of the relativistic harmonic oscillator
by the Infeld-Hull factorization method, Theor. Math. Phys., 56 (1984), pp. 563–572.
[8] N. M. ATAKISHIYEV, A. F RANK , AND K. B. W OLF , A simple difference realization of the Heisenberg
¹ -algebra, J. Math. Phys., 35 (1994), pp. 3253–3260.
[9] N. M. ATAKISHIYEV, E.I. JAFAROV, S.M. N AGIYEV, AND K.B. W OLF , Meixner oscillators, Revista Mexicana de Fı́sica, 44 (1998), pp. 235–244.
[10] N. M. ATAKISHIYEV, M. R AHMAN , AND S. K. S USLOV , On Classical Orthogonal Polynomials, Constr.
Approx., 11 (1995), pp. 181–226.
[11] N. M. ATAKISHIYEV AND S. K. S USLOV , Difference analogs of the harmonic oscillator, Theor. Math. Phys.,
85 (1991), pp. 442–444.
[12] N. M. ATAKISHIYEV AND S. K. S USLOV , A realization of the q-harmonic oscillator, Theor. Math. Phys., 87
(1991), pp. 1055–1062.
[13] N. M. ATAKISHIYEV AND B. W OLF , Approximation on a finite set of points through Kravchuk functions,
Revista Mexicana de Fı́sica, 40 (1994), pp. 1055–1062.
[14] G. B ANGEREZAKO , Discrete Darboux transformation for discrete polynomials of hypergeometric type, J.
Phys A: Math. Gen., 31 (1998), pp. 2191–2196.
[15] G. B ANGEREZAKO , The factorization method for the Askey–Wilson polynomials, J. Comput. Appl. Math.,
107 (1999), pp. 219–232.
[16] C. B ERG AND A. R UFFING , Generalized ¹ -Hermite polynomials, Comm. Math. Phys., 223 (2001), pp. 29–46.
[17] R. F LOREANINI , J. L E T OURNEUX , AND L. V INET , More on the ¹ -oscillator algebra and ¹ -orthogonal
polynomials. J. Phys. A: Math. Gen., 28 (1995), pp. L287–L293.
[18] R. F LOREANINI AND L. V INET , ¹ -orthogonal polynomials and the oscillator quantum group, Lett. Math.
Phys. 22 (1991), pp. 45–54.
[19] L. I NFELD AND T. E. H ULL , The factorization method. Rev. Modern Phys., 23 (1951), pp. 21–68.
[20] R. K OEKOEK AND R. F. S WARTTOUW , The Askey-scheme of hypergeometric orthogonal polynomials and
its q-analogue, Reports of the Faculty of Technical Mathematics and Informatics, No. 98-17, Delft University of Technology, Delft, 1998.
[21] A. L OREK , A. R UFFING , AND J. A. W ESS , A ¹ -deformation of the harmonic oscillator, Z. Phys. C, 74
(1997), pp. 369–377.
[22] M. L ORENTE , Raising and lowering operators, factorization method and differential/difference operators of
hypergeometric type, J. Phys. A: Math. Gen., 34 (2001), pp. 569–588.
[23] W. M ILLER (J R .), Lie theory and difference equations, I, J. Math. Anal. Appl., 28 (1969), pp. 383–399.
[24] W. M ILLER (J R .), Lie theory and ¹ -difference equations, SIAM J. Math. Anal., 1 (1970), pp. 171–188.
[25] S H . M. N AGIYEV , Difference Schrödinger equation and ¹ -oscillator model, Theor. Math. Phys., 102 (1995),
pp. 180–187.
[26] A. F. N IKIFOROV, S. K. S USLOV, AND V. B. U VAROV , Classical Orthogonal Polynomials of a Discrete
Variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
[27] A. F.N IKIFOROV AND V.B.U VAROV , The Special Functions of Mathematical Physics, Birkhäuser, Basel,
1988.
[28] E. S CHR ÖDINGER , Further studies on solving eigenvalue problems by factorization, Proc. Roy. Irish. Acad.,
A 46 (1941), pp. 183–206.
[29] Y U . F. S MIRNOV , Factorization method: New aspects, Rev. Mexicana Fı́s., 45 (1999), pp. 1–6.