ETNA
Electronic Transactions on Numerical Analysis.
Volume 27, pp. 140-155, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
AN ALGEBRA OF INTEGRAL OPERATORS
SERGEI K. SUSLOV
Abstract. We introduce an algebra of integral operators related to a model of the -harmonic oscillator and
investigate some of its properties.
Key words. integral operators, divided difference operators, the continuous -Hermite polynomials, generating
functions, Poisson kernel, bilinear generating functions, -harmonic oscillators
AMS subject classifications. 33D45, 42C10, 45E10
1. Introduction. In this report a unification of the basic analog of Fourier transform and
inverses of the Askey–Wilson divided difference operators will be given in a form of certain
algebraic structure related to a model of the -harmonic oscillator. We present here only the
summary of results; a paper with detailed proofs will appear elsewhere [53].
with
To be more specific, let us consider a -quadratic lattice of the form
and let us introduce the symmetric difference operator as
!"!#$% &'(#)+*, -/. &'(#0.1*, -2
The first order Askey–Wilson divided difference operator is given by
3"4 &56 & "(#)+*, 78/. "(#0.1*, 78
"9#:;*<.="!#>.1*, 7 2
3 4
3 4
Several
inverses @? of the Askey–Wilson divided difference operator, such that @?
3 4 +A “right”
and A is the identity operator, were constructed in [20, 31, 33],
It was Dick Askey who realized that Wiener’s treatment of the Fourier integrals [59]
contains the key to -extensions [8, 11, 41]. Generalizing Wiener’s method to the level of
(1.1)
the Askey–Wilson polynomials one can introduce a set of one-parameter integral operators
which resemble raising and lowering operators. These operators obey an interesting algebraic
structure which allows to obtain one-sided inverses of the divided difference operators of the
first order [20, 31, 33], and to find the resolvents of the second order Askey–Wilson operators
in different spaces of functions. The aim of the present note and its extended version [53] is to
consider the simplest case related to the continuous -Hermite polynomials; a more general
case including the Askey–Wilson polynomials will be discussed later; see also [49] and [50]
for an extension of the Askey–Wilson polynomials orthogonality to a certain class of
functions.
F
F
BDC<E
The paper is organized as follows. In 1 to 4 we remind the reader basic facts about the
continuous -Hermite polynomials and consider a model of -harmonic oscillator in terms
of these polynomials. In 5 to 7 we introduce a family of one parameter integral operators,
which extend rasing and lowering operators, and investigate some properties of these operators, their adjoints and inverses in a framework of a single algebraic structure. An analog
F
F
G Received May 30, 2003. Accepted for publication January 10, 2004. Recommended by F. Marcellán.
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287 (sks@asu.edu,
http://hahn.la.asu.edu/˜suslov/index.html). The author was supported by National Science
Foundation under contract DMS-9803443.
140
ETNA
Kent State University
etna@mcs.kent.edu
141
AN ALGEBRA OF INTEGRAL OPERATORS
F
of the -Fourier transform is briefly discussed in 8. An explicit realization of the number
operator in this model of the -oscillator terms of Hadamard’s principal values integral is
outlined in 9. Inverses of the first order Askey–Wilson operators are constructed in 10. In
conclusion, the resolvent and Green’s function of the corresponding -Hamiltonian are found
in 11. More details can be found in the forthcoming paper [53].
F
F
F
2. Continuous -Hermite Polynomials. Although the continuous -Hermite polynomials were originally introduced by Rogers [42], [43], [44], their orthogonality relation and
asymptotic properties had been established only recently by Allaway [2], Al-Salam and Chihara [4], and Askey and Ismail [9], [10]. These polynomials are given by
(2.1)
NI
HJIK/L M OQPSR UTV !O UT$U TVI I O W I YX OQZ ,[
\+]_^7`ba
and the continuous orthogonality relation is [2], [9], [10]
cd
R Hfef]_^7`baYL bHJI@!]^`bagL h9 XVi [ X$i TV jDkml7a
n,o !UUT$TV 8k I eI [
(2.2)
or
c ?
H I &<L bH e &/L qpYY'lr+l XI eI [
S?
(2.3)
where the weight function is
(2.4)
and the
k
w
?
V
v
X
psY%+t"h$*u. X j OQPSR $h *:1xh-*u.y, X jq O X O j
z X -norm is given by
(2.5)
l XI n,o ! U UTVT$ -k I 2
The continuous -Hermite polynomials (2.1) obey a very important property, namely, the
action of the Askey–Wilson divided difference operator (1.1) on
results in the same
polynomial of the lower degree,
H I <L I
HJIS&/L %n W{?| I Z Vv X *uu* .}.} HJI @? <L [
which is a -analog of the familiar formula HI ~ &€,@H I
@? & . It is worth also noting
that the continuous -Hermite polynomials are the simplest special case of the fundamental
Askey-Wilson polynomials ‚ I &qTVƒ [V„[V…,[ lU [14] corresponding to the zero-valued parameters,
H I &<L †‡‚ I qT$ˆ [ ˆ [ ˆ [ ˆ7 . They satisfy a second-order difference equation and have the
Rodrigues-type formula among other properties; see, for example, [5], [14], [16], [18], [30],
[32], [40], and [47] for more details.
ETNA
Kent State University
etna@mcs.kent.edu
142
SERGEI K. SUSLOV
3. Bilinear Generating Functions. There are several important generating functions
for the continuous -Hermite polynomials; see, for example, [6], [30], [48], and [51]. The
Poisson kernel of Rogers, or the -Mehler formula, is one of them
k I
N
‰
Š
& [-‹ 56 SP Rsl Œ IX HJIK&<L HJIK ‹ L (3.1)
I
o† V@i [ _h(  [ Œ [ XT$ ijŽk [ Yi_Yi TV k
Œ
Œ
Œ
Œ
with r;]_^7`ba [‹ ]^`C [ L LU‘’* and lXI defined by (2.5). A related kernel is
ΠNk I
Š
“
(3.2)
& [-‹ )5” PSR•l Œ XI H I <L H I –? ‹ L I
o† iV@i [ ‹ i. _YŒ i [ h i[  YŒ X i TV[ j k Yi T$ k 2
Œ
Œ
Œ
Œ
Both kernels (3.1) and (3.2) are special cases —J;ˆ and —†˜* [ respectively, of a more general
Carlitz’s formula,
(3.3)
where
Nk
I
O
PI @R (UŒT$ I HfIK<L HJI  ‹ L O TV j_k™(UT$ O ‚ O h!]_^7`C)T Ži [ 7Yi_j
h
Ž
X
Œ iV@ [ i_Yi [ iYi [ Œ _ Œ T$ k [
Œ
Œ
Œ
Œ
O
‚ O !]^7`C:TVƒ [V„ M ƒ !O ƒ („ UTVT$ O O†š C X"› [ ƒ ƒU„[  [ ˆ ƒ7 T– [ œ
are the Al-Salam and Chihara polynomials; see, for example, [14] and [34]. Carlitz [22] derived (3.3) using series manipulations; Al-Salam and Ismail [5] gave another proof using the
fact that the continuous -Hermite polynomials are the moments of the distribution function
of the Al-Salam and Carlitz polynomials [3].
Let us also consider another related kernel
k
e
N
Š

(3.4)
[-‹ 56 PSR H e <L bH e <? ‹ L l Xe Œ
ek
–?
N I “ Š 48ž [$‹
PSR I
‰ “  kernels as follows:
and introduce the generalizations of the [ and
k I
OW Z
N
“
Š
& [$‹ 5” PSRKl Œ XI HJI@&/L HfI  O ‹ L (3.5)
IO
O O
o†h [ Œ VX @ i T$[ j kn_!UT$ [ i‚ h ‹ [ T Œ Yi i[_Œ Y iY iTV_ j k
Œ
Œ
Œ
Œ
ETNA
Kent State University
etna@mcs.kent.edu
143
AN ALGEBRA OF INTEGRAL OPERATORS
and
 Š W QO Z [-‹ 56
Nk I
PSR I
(3.6)
Nk
O ‹ L e O
e
e
H
<
L
b
H
l Xe Œ 
e PSO R
h! TV j I “ ŠW 4OQž Z & [-‹ Q2
UTV I
With the help of these kernels (3.1), (3.2), (3.4)–(3.6) we shall introduce in this paper a family
of integral operators related to the so-called -Heisenberg algebra.
4. The -Heisenberg Algebra. Recent advances in quantum groups has led to a study
of the so-called -harmonic oscillators, originally introduced by Arik and Coon [7] and then
rediscovered by Biedenharn [19] and Macfarline [38]; see, for example, [12], [13], [17],
[27], [28], [29], [25], [26], [57], [61], and references therein. The -oscillator is a simple
quantum mechanical system described by an annihilation operator and a creation operator
parameterized by a parameter . The basic problem is to find realizations of these operators
as differential, difference or integral operators acting on appropriate functional spaces. The
first model of -oscillator in a Hilbert space of analytic functions was discussed in [7]. Later
introduced models of -oscillators are closely related to the -orthogonal polynomials. The
-analogs of boson operators have been studied by various authors and the corresponding
wave functions were constructed in terms of the continuous -Hermite polynomials of Rogers
[42]–[44] by Atakishiyev and Suslov [15] and by Floreanini and Vinet [29]; in terms of the
Stieltjes–Wigert polynomials [46], [60] by Atakishiyev and Suslov [17]; and in terms of
-Charlier polynomials of Al-Salam and Carlitz [3] by Askey and Suslov [12], [13] and by
Zhedanov [61]. The model related to the Rogers–Szegő polynomials [54] was investigated by
Macfarline [38] and by Floreanini and Vinet [27]. In this note we shall restrict ourselves only
to the model related to the continuous -Hermite polynomials where the weight function
given by (2.4) is continuous and positive on
and the corresponding wave functions
form a complete system.
are associated with the wave functions for the
Just as the Hermite polynomials
harmonic oscillator [36], the continuous -Hermite polynomials
are associated with
the normalized -wave function for the -harmonic oscillator,
psY
8.x* [ *Ž
H I &
H I &/L Ÿ I <L :¡ h I –?,T$ jŽk ?-vVXM£ Yp YH I <L [
o ¢
so that the orthogonality relation (2.2)–(2.5) now reads
c ? Ÿ
I &/L Ÿ I &<L l7\ eI 2
@?
4
4
The -annihilation operator ƒ Y and the -creation operator ƒ  &
tation rule
ƒ 4 Y-ƒ 4 &Y.= @? ƒ 4 &8ƒ 4 Y%¤*
that satisfy the commu-
were introduced explicitly in [15]. In this paper we shall consider another form of the -boson
operators which is equivalent to those given in [15], but more convenient for our purposes;
see [28].
ETNA
Kent State University
etna@mcs.kent.edu
144
SERGEI K. SUSLOV
The -annihilation operator
commutation rule
ƒf;ƒ 4 &
and the -creation operator
„ ƒb4 &
satisfy the
ƒ „ .= S? „ ƒ'¥*
(4.1)
and act on the corresponding -wave functions as follows
*u.} I œ ?$vVX L §.¨*Ž¦ [
*u.= @?
*u.} I @? œ ?$vVX L †;*Ž¦Y2
(4.3)
*u.} S?
k
Let © be a space of analytic functions spanned by ªDH I <L |«L I PSR and let the weighted inner
product in © be
c ?
–
[
­
!¬ $®x56
¬ Y ­ &YpsYl7 [
(4.4)
S?
where ¯ denotes the complex conjugate; we need also impose certain analyticity condition on
the functions ¬ and ­ [53]; see also [52], [45] for the maximum domain of analyticity of the
series in the continuous -Hermite polynomials.
In this paper we consider the following explicit realization of the -annihilation ƒ and
-creation „ operators. One can easily verify that the divided difference operators
(4.5)
ƒf¤. -*u.}?-vVX ?$vVX h .} j @?0° ?$vVX± .} @?-vVX/±² [
„ ¥. * ?-vVX h .= S
j S? ° YX|
?$v$X/± .} X|
@?-vVX±7² [
(4.6)
-*.} ƒKL q¦M ›
„ L q¦M ›
(4.2)
acting on analytic functions of the form
where
³_´bµK(¶r·S
¬s(#Ž: Ÿ –!#$ [
–(#Ž: h j [
is the shift operator,
³_´bµK(¶r·SU¬s(#ŽM;¬s(#)¶/ [
indeed, satisfy the -commutation rule (4.1). Moreover, it is easy to see that these operators
are adjoint to each other,
„ ¬ [-­ ® ¥!¬ [ ƒ ­ ® [
with respect to the inner product (4.4) in the space of analytic functions under consideration.
‰ “

5. Introducing Integral Operators. Using the , and
kernels given by (3.1)–
(3.2), (3.4) for
let us consider the following integral operators
(5.1)
(5.2)
L UL ‘’* [
Œ
c
¹ ¸ U ¬•&: ? ‰Š & [-‹ U¬• ‹ Up ‹ bl ‹Y[
Œ
c S? ? “%Š
b¬s&%
& [$‹ U¬s ‹ bpY ‹ l ‹[
S?
Œ
ETNA
Kent State University
etna@mcs.kent.edu
AN ALGEBRA OF INTEGRAL OPERATORS
(5.3)
º U ¬•&% c ?  Š Y‹ [ Y U¬• ‹ Up ‹ bl ‹ 2
S?
Œ
and the corresponding adjoint operators
145
c
¼ » U ¬•&: ? “%Š ‹[ Yb¬s ‹ Up ‹ bl ‹[
Œ
c S? ?  Š
(5.5)
U¥&:
[-‹ U¬• ‹ Up ‹ bl ‹ 2
S
?
Œ
¼
¹
with respect to the inner product (4.4). Indeed,
¬ [$­ 8®K¥!¬ [ » ­ 8® [ º ¬ [$­ 8®•¤¬ [ ­ $ ®
by the Fubini theorem when L Lx‘½*T see, for example, [1], [23], [35], [37], [56] for an
extensive theory of the integral Πoperators.
¹ setting, let us introduce also
In a more general
c
W OQZ U¬sY% ? “ ŠW OQZ [$‹ b¬s ‹ Up ‹ bl ‹[
S?
» W OQZ Œ U¬sY% c ? “ ŠW OZ ‹[ U¬s ‹ bpY ‹ l ‹[
@?
º¼ W OQZ Œ U¬•&Y: c ?  Š W OQZ ‹[ Yb¬s ‹ Up ‹ bl ‹[
OW QZ Œ U¬sY% c @? ?  Š W OQZ [-‹ U¬• ‹ Up ‹ bl ‹
@?
Œ
and with the help of (3.6) verify that
k
º W OQZ M N I h O T$ j I » W OQZ I [
¼
¹
ΠI k P@R UO TV 8I
Œ
OW QZ : N I h! TV j I W OQZ I S2
Œ
ΠI PSR UTV I
¹
Once again,
° W OQZ ¬ [-­ ² ® ° ¬ [ ¼» W OQZ ­ ² ® [
° º W OQZ ¬ [-­ ² ® ° ¬ [ W OQZ ­ ² ® [
in the space of analytic functions, if L LU‘’*2
¹ that
It is easy to show
Œ
W OZ -HfeJ<L % e O !U!UT$ T$ e e O HJe O &/L [
(5.6)
» W OQZ Œ -H e <L : Œ e H e O &/L [
(5.7)

Q
O
Z
Œ
Œ
º
U
e
W 8HJe'<L : UTV TV e 8e O HJe  O <L [
(5.8)

Œ
Œ
(5.4)
ETNA
Kent State University
etna@mcs.kent.edu
146
¼
SERGEI K. SUSLOV
W OQZ 8 HJeJ&<L % e O HJe O &/L [¿¾ÁÀ —2
Œ (5.6)–(5.9) define
Πa set of integral operators that correspond to the
For —\Â* these relations
¹
so-called raising and lowering operators for the continuous -Hermite polynomials:
(5.10)
-H e <L % e @? 8*u.} e bH e @? <L [
»
(5.11)
Œ 8H e &<L % Œ e H e e –? <L [
¼º Œ 8H e &/L % Œ
(5.12)
e <? H e <? &/L [
u
*
}
.
Œ
à ˆ2
Œ 8HJeJ&<L % e S? HJe S? &/L [¿¾¡
(5.13)
Œ
Œ
¸
The continuous -Hermite polynomials are eigenfunctions of the -operator:
¸ -HJeJ&<L % e HJe'<L Q2
(5.14)
Œ
Œ
¹
Combining (5.10) and (5.11) we find
» bHJe'<L :¤ e @? 8*u.} e HJe'<L Q2
(5.15)
? ŒX
?ŒX
Œ
Œ
This integral equation with two free parameters and extends the corresponding second
X
order difference equation for the continuous -Hermite
see
¼ [40]
¹ [14], [18] and
Œ ? Œpolynomials;
for more details on this equation. Another integral equation follows from (5.12)–(5.13).
¸ ¼ » º
¹
6. “Algebra” of Integral Operators. The integral operators [ [ [ [ and obey
the following multiplication rules:
º » ¸ ¹
X
X
X
X
X
¸ Eq. Œ (6.1)
Eq. Π(6.3)
Eq.Π(6.5)
Eq. Π(6.7)
Eq. Π(6.9)
?
(6.2) Eq. (6.16) Eq. (6.12) Eq. (6.10) Eq. (6.18)
¼» Œ Œ ? Eq.
(6.4) Eq. (6.13) Eq. (6.17) Eq. (6.20) Eq. (6.11)
º Œ ? Eq.
Eq. (6.6) Eq. (6.11) Eq. (6.21) Eq. (6.22) Eq. (6.14)
Π?? Eq. (6.8) Eq. (6.19) Eq. (6.10) Eq. (6.15) Eq. (6.23)
All the products
Πin this table can be evaluated directly from the definitions of the integral
operators and corresponding kernels in the following manner:
¹¸ ¸
¹
(6.1)
? ¹ ¸ X M ¸ ¹ ? X [
(6.2)
Π? ΠX M X ΠΠ? X [
¸
Œ ? ¸ Œ X M »Œ ? Œ X Œ [
(6.3)
»
Œ ? » Œ X % » Œ ? Œ X [
(6.4)
¸
Œ ? ¸ Œ X % º ? Œ Œ ? X [
(6.5)
º
(6.6)
¼¸ Œ ? º Œ X % Œ º¼ ? Œ X Œ [
(6.7)
Œ ? ¼ ¸ Œ X % ¼ ? Œ Œ ? X [
(6.8)
¹¸ Œ ? Œ X M Œ ¼ X Œ ? Œ X [
(6.9)
Œ ? ¹ º Œ X M Œ ? Œ X ¼ » Œ [
(6.10)
¹º Œ ? Œ X % » Œ ? Œ X M ¸ ? X [ ¸
Œ ? ¹» Œ X % ¸ Œ ? Œ X M¸ ¤ ? Œ X Œ S? ? X . ¸ !ˆ7$ [
(6.11)
Œ Œ
Π? ΠX M Π? X /.}ΠΠ? ΠX [
(6.12)
»
¸
¸
(6.13)
Œ ? Œ X MÄ ? Œ X Œ @? ? X Œ /.Œ ! ? X - [
Œ
Œ
Œ Œ
Œ Œ
Œ Œ
(5.9)
ETNA
Kent State University
etna@mcs.kent.edu
¼
147
AN ALGEBRA OF INTEGRAL OPERATORS
k
N
X MÄ ? X S?Å OPSR ¸ h ? X O j . ¸ !ˆ7ÇÆ [
Œ
Œ Œ
Œ Œ
Nk ¹ ¸
O
O
X M OQPSR h ? X jq [
Œ
Œ Œ
X % X WÈX Z Z ? X [
Œ X M Œ ? » k WÈX Œ ? ¹ Œ X [
Œ Œ N Œ Œ
X % X OQPSR X O ¹ WÈX Z h ? X O j [
ΠΠk
Œ Œ
N O
X % X OQPSR h ? X O j [
ΠΠk
Œ Œ
N O » WÈX Z
h ? XOj [
X M ? OQPSR ΠΠk
Œ Œ
N X O » WÈX Z h O j [
X M ? OQPSR ?
X
ΠΠk
Œ Œ
N O *u.} O –? » WX Z
X M ? OQP@R *u.= ¹ h ? X O j [
Œ Œ
ΠΠk
N O *u.} O –? WÈX Z h O j
X % ? OQPSR *u.}
Œ Œ
Œ?ŒX
* [ when all integral operators are bounded.
?Œ L [ L Œ X ofL )‘n
Although this “algebra”
integral operators is not closed, it unifies many important
º
(6.14)
¼ ?
Œ
¹ ? ¹º
(6.15)
Œ
?
(6.16)
»¹ Œ » ¼
(6.17)
Œ?
(6.18)
¼ ?¹
Œ
?
(6.19)
Œ
»
(6.20)
? º
Œ
º
(6.21)
?»
Œ
º
º
(6.22)
¼ ? ¼
Œ
(6.23)
?
Œ
and so on. Here É†Ê ´•$L
properties of these operators in a single algebraic structure and deserves detailed study. For
instance, it contains the inverses of the Askey–Wilson divided difference operators [20] and
the -Fourier transform [8] as special cases after certain analytic continuation with respect to
the free parameter. We shall consider several important examples.
7. Some Degenerate Cases of Integral Operators. So far we have considered the integral operators (5.1)–(5.5) with
, when they are bounded. In this section we shall
consider analytic continuation of these integral operators outside the interval
This
leads to several important (unbounded) operators, when
etc.
L )L ‘Ë*
Œ
7.1. Operator
or
¸ 8*Ž
¥* [ S?$vVX [ S? [
¸ 8* . It can be shown that
Œ
c
ŠÏÍÈÎ É ? ‰Š [$‹ ¬Ñ ‹ Up• ‹ xl ‹ ;¬Ò& [
?QÐ S?
is the identity operator
ˆJÌ ’
‘ *2
Œ
¸ $*ŽM;A
in the space of analytic functions under consideration; see [53] for more details.
7.2. Operator
(7.1)
¸ h @?-vVX j . In a similar fashion
¸ ° @?-vVX ² @?-vVX/±x.}
| ?$v$X± 2
.} ETNA
Kent State University
etna@mcs.kent.edu
148
SERGEI K. SUSLOV
¸ h O vVX j
¸ h @?-vVX j
This can be shows as a result of “collision” of the poles in the complex plane [53].
Now operators
can be found as products of
from (7.1)
For example,
A
O
° ¸ ° @?-vVX ²² ¸ ° O vVX ² 2
5
¸ h S? j ¸ ° S?$v$XD² ¸ ° @?-vVXD²
! .} Yh(X| V
ÇS–?$?$vVvVX:X ,.=± S
VS?$vVX j
.} ,X|h(
V S
V?$@vV?-X:vV7X .}± Ǐ<?-vVX j
. h |S?$v$X .= S
Ǐ–,?$X|v$
VX @j0?-h vV X
Ǐ–A ?$vVX .} V@?-vVX j [
¸ h9@?j
where is the identity operator.
The operator
is closely related to the Hamiltonian of the model of the -harmonic
oscillator under consideration, namely,
¸
HÓ „ ƒf u* . u* .} h SS? ? j
¸
¸
h *u. h$*u.=h S@?$vV?-XvVjŽX j0j0hh$*:*:Ô @h?-vVXSj ?$vVX jŽj 2
Here ƒ and „ are the ¹ -annihilation and -creation operators, given by (4.5) and (4.6), respectively.
»
7.3. Operators h(S?$v$XŽj and h(S?$vVXDj . “Colliding” the poles in the complex plane
¹
[53], one can show that
° @?-vVX_² ¥$*.= ?$v$X ƒ [
which is up to a factor just the first order Askey–Wilson divided difference operator; cf. (1.1)
and (4.5).
In a similar manner,
» ° S?$v$X ² ¤8*u.} ?$vVX „ 2
¹
¹
From the multiplication table of the integral operators we obtain the following ( )commutators:
¹ ? » X /. ? X » X ¹ ? %¥-*u.= ¸ ? X [
Œ Œ Œ Œ Œ Œ
Œ Œ
»
»
¸
? X /.= ? X X ? %¥$*.= ? X 2
Œ Œ
Œ Œ Œ Œ
Œ Œ
The special cases ?ΠΠX S?$v$X are well-known in the theory of -oscillators [19], [38]:
ƒ „ .} S? „ ƒf;A [ ƒ „ . „ ƒf ¸ h @? j 2
ETNA
Kent State University
etna@mcs.kent.edu
149
AN ALGEBRA OF INTEGRAL OPERATORS
¹
¹
Also, from the multiplication table of the integral operators,
? ¸ X M X ¸ X ? [
Œ Œ Œ Œ Œ
» ¸ M S? ¸ » [
? ŒX ŒX ŒX Œ?
Œ
and when S?$v$X [
?Œ
XΠΠone gets
ƒ ¸ % ¸ bƒ [
Œ Œ Œ
¸„ M S? ¸ „ 2
Œ Œ Œ
In the case €US?$vVX we can use these relations in order to determine the spectrum of the
-Hamiltonian
ΠH in a pure algebraic form.
The normalized -wave functions in the model of -oscillator under consideration are
¬ I !YMÓÕ h I –? TV jDk oÖ ?-vVX H I /L with the orthogonality relation
c ?
¬ I U¬ e &p•&blr e)I
@?
and the explicit action of the -annihilation and -creation operators on these wave functions
is
I
ƒ¬ I › *u*u.}.= @? œ ?$vVX ¬ I S? [
I
„ ¬ I › *u*u.}.} S@? ? œ ?$vVX ¬ I –?
according to the general rule (4.2)–(4.3).
8. Generalized -Fourier Transform and its Inverse. The semi-group property from
the multiplication table is
¸ ¸ M ¸ [
? ŒX
?ŒX
Œ
Œ
¸
when ɆÊ,´x$L L [ L L "‘€*2 “Analytic continuation” of the integral operators the
?
X
?|× X onin the
*
unit circle L Œ LqÓ
results in the -Fourier transform [8], [11], [41], [53] (usually,
Œ
Œ
classical case,
Œ ?|× X ØÒ¤o– [55], [59], but we discuss the general case with ˆ§‘’Ø}‘o ). Then,
formally,
¸ h( Ù j ¸ h! ÈÙ j>;A [
A
where is the identity operator. The explicit transformation formulas in the spaces of analytic
functions can be given in terms of Cauchy’s principal value integral. The -Fourier transform
and its inverse are certain singular integral equations, somewhat similar to the case of the
classical Hilbert transform; see [24], [39] for an account of the theory of singular integral
equations.
ETNA
Kent State University
etna@mcs.kent.edu
150
SERGEI K. SUSLOV
9. The “Number” Operator. The concept of number operator is well-known in quantum mechanics [36]. Similar operators were formally introduced in the theory of -harmonic
operators [19], [38], but explicit realizations of these “number” operators were not constructed. In the model of the -oscillator under consideration it is natural to introduce this
operator as the generator of the semi-group of the integral operators
[53]. Denote
¸ Œ
¸"Ú ¸ _L Š qP Û8Ü [
Œ
or in the form of a contour integral,
¸"Ú ¬ÒY% cÝ ‰ Û Ü & [-‹ x¬Ò ‹ xps ‹ bl ‹[
¸
where Þ is a contour corresponding to analytic continuation of the operator to the values
L LUßn*7T see [53] for the details. Then the semi-group properties can be writtenŒ as usual
Œ
¸ R +A [ ¸ Ú ¸Kà ¸ Ú à

and formally
k
¸ Ú á³_´Uµ!¶gâ:% N ! ¶gâ% I [
I SP R  ã
where by the definition the “infinitesimal” operator is
¸'Ú
l
âá56 › l7¶ åœ ää Ú P@R 2
ää
â
Explicit realization of the number operator
can
be
given in terms
¹
¹ of Hadamard’s principal
value integral in the space of analytic functions; see [53] for the details.
»
»
10. Inversions of Operators and . The operators and are bounded
integral operators for L Lb‘n*7T they admit
Πan analytic
Πcontinuation in theΠlarger domain
Œ L Lbß*
and become unbounded
Œ divided difference operators when +@?-vVX2 The problem of finding
Œ
inverses of these operators is similar to the familiar classical
Πresults
l c &Y"lr & [
lc l &Y"lr &q constant 2
l
¹
¼
In the model of the -harmonic oscillator under consideration we can extend these relations to
» -derivatives, or even to -“fractional” derivatives,
our integral operators and
º namely,
, with the help of the integral
2 Indeed, for L LMԾ* oneΠcan
operators
and
¹ of the operators
write
that Œ
Πfrom the table of multiplication
Œ
Œ
¼
º
h S? j M ¸ 8*ŽMA [
¼
Œ » h S? Œ j ¸ 8*ŽM¹ A [
Œ Œ
º
where A is the identity operator. Thus the bounded integral operator ( ) with L LU‘*
»
gives the right (left) inverse of the unbounded operator h @?_j ( h SΠ?Dj ) [ provided
that
Œ
Πthis
operator is properly analytically continued to the domain Œ @? ߀*Œ 2 When
S
$
?
V
v
X one
ää Œ ää
Œxç
ETNA
Kent State University
etna@mcs.kent.edu
151
AN ALGEBRA OF INTEGRAL OPERATORS
gets, as a special case, the right inverse of the Askey–Wilson first order divided difference
originally found by Brown and Ismail [20] in this model of -oscillator.
operator
In a similar manner,
¹
º h¼ S ? j> ¸ $ *Ž<. ¸ (ˆ7 [
» h Œ S? j Œ % ¸ $ *Ž<. ¸ (ˆ7 [
Œ
Œ
when L LU‘n* and
¹
¹
Œ
¼
¼
º
º
? » X /. ? X » X ? % ¸¸ !ˆ [
Œ X Œ ? /. Œ ? Œ X Œ ? Œ X % !ˆ [
Œ Œ Œ Œ Œ
Œ
i.e., these “commutators” act on a vector ¬ as the projection operator to the “vacuum” vector
¬ R 2 Indeed,
¸ (ˆ7¬M c ? ‰ R & [-‹ U¬Ò ‹ xps ‹ bl ‹
@?
¤¬ R [ ¬8®§¬ R [
where ¬ R ;l R @? H R <L is the “vacuum” vector.
11. Resolvents and Green’s Functions. The continuous -Hermite polynomials, or the
wave functions in the model of the -harmonic oscillator under consideration, satisfy two
difference equations. We derive corresponding resolvents and Green’s function.
11.1. First difference operator. Let us start from the following difference equation for
the continuous -Hermite polynomials
¸ ° @?-vVXŽ² H I &/L :; I vVX H I &<L [
which is the special case +U@?-vVX of (5.14) due to (7.1), and consider
Œ
° ¸ ° S?$vVX_² .}ègA ² ¬1 ­
(11.1)
é§ê
é§ê
with the resolvent
° ¸ ° S?$vVX ² .}ègA ² @? [ ¬1 ­ 2
¸
Multiplying (11.1) by the corresponding bounded integral operator h(?$v$XŽj [ one gets
° Ax.yè ¸ ° ?-vVXD²S² ¬1 ¸ ° ?-vVXD²>­:[
where A is the identity operator. Thus,
k O
O
N
¸
S
?
° A".}è ° ?$v$X ²S² è ° ¸ ° ?$vVX ²S²
OQPSR
N k O ¸ ° O v$X ²
OQPSR è ETNA
Kent State University
etna@mcs.kent.edu
152
by (6.1), and
SERGEI K. SUSLOV
é§ê
k O
N
¸
@
?
° ° S?$vVX ² .}ègA ² O PSR è ¸ ° W O –? Z v$X ² 2
é§ê
ê
c ?
­ Y%
& [-‹ ­ ‹ Kps ‹ l ‹
@?ë
with the kernel
ê
k O
N
! [-‹ M OQPSR è ‰ 4Vì6í$îï!ðÈñ9ò & [$‹ ë
N k HfIK<L HJIK ‹ L I vVX
PSR l XI h *u.yè I vVX j 2
I
I
Introducing the eigenvalues è I ; vVX and the orthonormal eigenfunctions ¬ I ;lI S? H I <L [
one can finally write
ê
k
N
& [-‹ % PSR ¬ I èg!YIfU.}¬ I è ‹ 2
I
ë
é\ê7éåô é§ê é§ô
The resolvent identity holds
!ó\.}èY
. [
(11.2)
So, the resolvent is an integral operator
see [1], [23], [35], [37] for more properties of the resolvent.
11.2. Second difference operator. Let us factor, first of all, the corresponding Askey–
Wilson difference equation of the second order (or the -Hamiltonian) in the following manner. At the level of the integral operators in Eq. (5.15) we have
¹
» %¥ @? ¸ /. ¸ ( -
? ŒX Œ?ŒX
?ŒX
?ŒX
Œ
Œ
Œ
¹
and, hence, for ?ΠΠX S?$vVX [
S? » ° S?$vVXD² ° @?-vVX_² ¸ h( S? j.}A2
¹
Therefore, instead of solving
° S? » ° S?$vVX ² ° S?$vVX ² è ² ¬¨ ­:[
one can solve a simpler equation
h ¸ h( S? j0.ó@AUjS¬1 ­%[ óÑ¥*u.yè
with the help of the resolvent
é ô
é ô
h ¸ h S? j .ó@A j S? [ ¬1 ­ 2
(11.3)
ETNA
Kent State University
etna@mcs.kent.edu
AN ALGEBRA OF INTEGRAL OPERATORS
Multiplying (11.3) by
¸ ! [
A
!A.=ó ¸ $b¬1 ¸ ( ­:[
where is the identity operator, and once again
k
N
¸
A.ó ( - S? OQPSR ó
Nk
OQPSR ó
é ô
or
153
O ¸ ( - O
O ¸ h! O j [
k O O
N
¸
@
?
h h @? j .=ó@A j OQP@R ó ¸ h –? j 2
This consideration gives an explicit representation for the resolvent in terms of the integral
operator
é ô
ô
c ?
­ Y%
& [-‹ ­ ‹ ps ‹ l ‹
S? ë
ô
k O
N
[-‹ M OPSR ó ‰ 4 í-îUï [-‹ ë
N k H I &/L bH I ‹ L I
PSR l XI -*u.=ó@ I 2
I
with the kernel
The resolvent identity (11.2) holds.
11.3. Green’s function. Let
where
‹ ô
is the Dirac delta function. Then
ô
h¸
ö &
See [53] for more details.
ô
&õ.= ~ M ! [ ~ Ups ~ [
ë
ô
h S? j .ó@A j ö [-‹ % &õ. ‹ ô
[-‹ % k & [-‹ Up• ‹ ëN
PSR H I l XI&/8L *u .=H ó@I I ‹ L I ps ‹ Y2
I
ö & [ ~ M
or
with
é ô
­ &õ.= ~ [
ETNA
Kent State University
etna@mcs.kent.edu
154
SERGEI K. SUSLOV
Acknowledgement. This work was completed when the author visited Department of
Mathematics and Statistics at Carleton University, Ottawa, Canada. The author thanks Mizan
Rahman for his hospitality and help. The author is grateful to the organizers of Bexbach’s
meeting for their invitation and hospitality.
REFERENCES
[1] N. I. A KHIEZER AND I. M. G LAZMAN , Theory of Linear Operators in Hilbert Space, Dover, New York,
1993.
[2] W. R. A LLAWAY , Some properties of the -Hermite polynomials, Canad. J. Math., 32 (1980), pp. 686–694.
[3] W. A. A L -S ALAM AND L. C ARLITZ , Some orthogonal -polynomials, Math. Nachr., 30 (1965), pp. 47–61.
[4] W. A. A L -S ALAM AND T. S. C HIHARA , Convolutions of orthogonal polynomials, SIAM J. Math. Anal.,, 7
(1976), pp. 16–28.
[5] W. A. A L -S ALAM AND M. E. H. I SMAIL , -Beta integrals and the -Hermite polynomials, Pacific J. Math.,
135 (1988), pp. 209–221.
[6] G. E. A NDREWS , R. A. A SKEY, AND R. R OY , Special Functions, Cambridge University Press, Cambridge,
1999.
[7] M. A RIK AND D. D. C OON , Hilbert spaces of analytic functions and generalized coherent states, J. Math,
Phys., 17 (1976), pp. 524–527.
[8] R. A. A SKEY, N. M. ATAKISHIYEV, AND S. K. S USLOV , An analog of the Fourier transformation for the
-harmonic oscillator, in Symmetries in Science, VI, Plenum, New York, 1993, pp. 57–63.
[9] R. A. A SKEY AND M. E. H. I SMAIL , A generalization of ultraspherical polynomials, in Studies in Pure
Mathematics, P. Erdös, ed., Birkhäuser, Boston, Mass., 1983, pp. 55–78.
[10] R. A. A SKEY AND M. E. H. I SMAIL , Recurrence relations, continued fractions and orthogonal polynomials,
Mem. Amer. Math. Soc., 49 (1984), pp. iv+108.
[11] R. A. A SKEY, M. R AHMAN , AND S. K. S USLOV , On a general -Fourier transformation with nonsymmetric
kernels, J. Comp. Appl. Math., 68 (1996), pp. 25–55.
[12] R. A. A SKEY AND S. K. S USLOV , The -harmonic oscillator and an analogue of the Charlier polynomials,
J. Phys. A., 26 (1993), pp. L693–L698.
[13] R. A. A SKEY AND S. K. S USLOV , The -harmonic oscillator and the Al-Salam and Carlitz polynomials,
Lett. Math. Phys., 29 (1993), pp. 123–132.
[14] R. A. A SKEY AND J. A. W ILSON , Some basic hypergeometric orthogonal polynomials that generalize Jacobi
polynomials, Mem. Amer. Math. Soc., 54 (1985), pp. iv+55.
[15] N. M. ATATISHIYEV AND S. K. S USLOV , Difference analogs of the harmonic oscillator, Theoret. and Math.
Phys., 85 (1990), pp. 1055–1062.
[16] N. M. ATAKISHIYEV AND S. K. S USLOV , Difference hypergeometric functions, in Progress in Approximation Theory, Springer Series in Computational Mathematics, 19, Springer–Verlag, New York, 1992,
pp. 1–35.
[17] N. M. ATATISHIYEV AND S. K. S USLOV , A realization of the -harmonic oscillator, Theoret. and Math.
Phys., 87 (1990), pp. 442–444.
[18] N. M. ATAKISHIYEV AND S. K. S USLOV , On the Askey–Wilson polynomials, Constr. Approx., 8 (1992),
pp. 363–369.
[19] L. C. B IEDENHARN , The quantum group
and a -analogue of the boson operators, J. Phys. Math.
A., 123 (1989), pp. L873–L878.
[20] B. M. B ROWN AND M. E. H. I SMAIL , A right inverse of the Askey–Wislon operator, Proc. Amer. Math. Soc.,
123 (1995), pp. 2071–2079.
[21] J. B USTOZ AND S. K. S USLOV , Basic analog of Fourier series on a -quadratic grid, Methods Appl. Anal.,
5 (1998), pp. 1–38.
[22] L. C ARLITZ , Generating functions for certain -orthogonal polynomials, Collect. Math., 23 (1972), pp. 91–
104.
[23] R. C OURANT AND D. H ILBERT , Methods of Mathematical Physics, John Wiley, New York, 1937.
[24] R. E STRADA AND R. P. K ANWAL , Singular Integral Equations, Birkhäuser, Boston, 2000.
[25] R. F LOREANINI , J. L E T OURNEUX , AND L. V INET , An algebraic interpretation of the continuous big Hermite polynomials, J. Math. Phys., 36 (1995), pp. 5091–5097.
[26] R. F LOREANINI , J. L E T OURNEUX , AND L. V INET , More on the -oscillator algebra and -orthogonal
polynomials, J. Phys. A., 28 (1995), pp. L287–L293.
[27] R. F LOREANINI AND L. V INET , -Orthogonal polynomials and the oscillators quantum group, Lett. Math.
Phys., 22 (1991), pp. 45–54.
[28] R. F LOREANINI AND L. V INET , Automorphisms of the -oscillator algebra and basic orthogonal polynomials, Phys. Lett. A, 180 (1993), pp. 393–401.
÷gøgùDúiûü
ETNA
Kent State University
etna@mcs.kent.edu
155
AN ALGEBRA OF INTEGRAL OPERATORS
[29] R. F LOREANINI AND L. V INET , A model for the continuous -ultraspherical polynomials, J. Math. Phys.,
36 (1995), pp. 3800–3813.
[30] G. G ASPER AND M. R AHMAN , Basic Hypergeometric Series, Cambridge University Press, Cambridge,
1990.
[31] M. E. H. I SMAIL , M. R AHMAN , AND R. Z HANG , Diagonalization of certain integral operators II, J. Comp.
Appl. Math., 68 (1996), pp. 163–196.
[32] M. E. H. I SMAIL , D. S TANTON , AND G. V IENNOT , The combinatorics of the Askey–Wilson integral, J.
Comp. Appl. Math., 68 (1996), pp. 163–196.
[33] M. E. H. I SMAIL AND R. Z HANG , Diagonalization of certain integral operators, Adv. Math., 108 (1994),
pp. 1–33.
[34] R. K OEKOEK AND R. F. S WARTTOW , The Askey-scheme of hypergeometric orthogonal polynomials and
its -analogues, Report 98–17, Delft University of Technology, Faculty of Information Technology and
Systems, Department of Technical Mathematics and Informatics, 1998; electronic version accessible at
http://fa.its.tudelft.nl/˜koekoek/askey.html.
[35] A. N. K OLMOGOROV AND S. V. F OMIN , Introductory Real Analysis, Dover, New York, 1970.
[36] L. D. L ANDAU AND E. M. L IFSCHITZ , Quantum Mechanics: Non-relativistic Theory, Addison–Wesley,
1998.
[37] P. D. L AX , Functional Analysis, John Wiley, New York, 2002.
[38] A. J. M ACFARLANE , On -analogues of the quantum harmonic oscillator and the quantum group
,
J. Phys. A., 22 (1989), pp. 4581–4588.
[39] N. I. M USKHELISHVILI, Singular Integral Equations, Groningen–Holland, 1953.
[40] A. F. N IKIFOROV, S. K. S USLOV, AND V. B. U VAROV , Classical Orthogonal Polynomials of a Discrete
Variable, Springer–Verlag, Berlin, 1991.
[41] M. R AHMAN AND S. K. S USLOV , Singular analogue of the Fourier transformation for the Askey–Wilson
polynomials, in Symmetries and Integrability of Difference Equations, CRM Proceedings and Lecture
Notes, 9, Amer. Math. Soc., Providence, RI, 1996, pp. 289–302.
[42] L. J. R OGERS , Second memoir on the expansion of certain infinite products, in Proc. London Math. Soc., 25
(1894), pp. 318–343.
[43] L. J. R OGERS , Third memoir on the expansion of certain infinite products, in Proc. London Math. Soc., 26
(1895), pp. 15–32.
[44] L. J. R OGERS , On two theorems of combinatory analysis and some allied identities, in Proc. London Math.
Soc., 16 (1917), pp. 315–336.
[45] M. S IMON AND S. K. S USLOV , Expansion of analytic functions in -orthogonal polynomials, J. Funct. Anal.
Approx. Theory, to appear.
[46] T. J. S TIELTJES , Recherches Sur Les Fractions Continues, Annales de la Faculté des Sciences de Toulouse,
No. 9, 1895.
[47] S. K. S USLOV , The theory of difference analogues of special functions of hypergeometric type, Russian Math.
Surveys, 44 (1989), pp. 227–278.
[48] S. K. S USLOV , “Addition” theorems for some -exponential and -trigonometric functions, Methods Appl.
Anal., 4 (1997), pp. 11–32.
[49] S. K. S USLOV , Some orthogonal very-well-poised
-functions, J. Phys. A., 30 (1997), pp. 5877–5885.
[50] S. K. S USLOV , Some orthogonal very-well-poised
-functions that generalize Askey–Wilson polynomials,
Ramanujan J., 5 (2001), pp. 183–218.
[51] S. K. S USLOV , An introduction to basic Fourier series, Developments in Mathematics, Vol. 9, Kluwer Academic, Dordrecht, 2003.
[52] S. K. S USLOV , An analog of the Cauchy–Hadamard formula for expansions in -polynomials, in Theory and
Applications of Special Functions, M. E. H. Ismail and Erik Koelink, eds., Developments in Mathematics, Vol. 13, Springer, New York, 2005, pp. 443-460. Â
[53] S. K. S USLOV , An algebra of integral operators in a model of -harmonic oscillator, under preparation.
[54] G. S ZEG Ő , Beitrag zur Theorie der Thetafunktionen, Sitz. Preuss. Akad. Phys. Math. Kl., XIX (1926),
pp. 242–252; reprinted in Collected Papers, Vol. I, R. Askey, ed., Birkhauser, Boston, 1982.
[55] E. C. T ITCHMARCH , Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford University Press, 1948.
[56] F. G. T RICOMI , Integral Equations, Dover, New York, 1985.
[57] J. VAN DER J EUGHT , The -boson operator algebra and -Hermite polynomials, Lett. Math. Phys., 24
(1992), pp. 267–274.
[58] E. T. W HITTAKER AND G. N. WATSON , A Course of Modern Analysis, Cambridge University Press, Cambridge, 1952.
[59] N. W IENER , The Fourier Integral and Certain of Its Applications, Cambridge University Press, Cambridge,
1933.
[60] S. W IGERT , A, Arkiv för Matematik, Astronomi och Fysik, 17 (1923), pp. 1–15.
[61] A. Z HEDANOV , Weyl shift of -oscillator and -polynomials, Theoret. and Math. Phys., 94 (1993), pp. 219–
224.
÷Yø%úûü(ù
ýQþÿ
ýQþÿ