Printer Friendly
The Free Library
22,741,889 articles and books

An algebra of integral operators.



Abstract. We introduce an algebra of integral operators related to a model of the q-harmonic oscillator oscillator

Mechanical or electronic device that produces a back-and-forth periodic motion. A pendulum is a simple mechanical oscillator that swings with a constant amplitude, requiring the addition of energy at each swing only to compensate for the energy lost because of air
 and investigate some of its properties.

Key words. integral operators, divided difference operators, the continuous q-Hermite polynomials, generating functions, Poisson kernel In potential theory, the Poisson kernel is the derivative of the Green's function for the two-dimensional Laplace equation, under circular symmetry, using Dirichlet boundary conditions. It is used for solving the two-dimensional Dirichlet problem. , bilinear bi·lin·e·ar  
adj.
Linear with respect to each of two variables or positions. Used of functions or equations.

Adj. 1. bilinear - linear with respect to each of two variables or positions
 generating functions, q-harmonic oscillators

AMS AMS - Andrew Message System  subject classifications. 33D45, 42C10, 45E10

1. Introduction. In this report a unification of the basic analog of Fourier transform Fourier transform

In mathematical analysis, an integral transform useful in solving certain types of partial differential equations. A function's Fourier transform is derived by integrating the product of the function and a kernel function (an exponential function raised to
 and inverses of the Askey-Wilson divided difference operators will be given in a form of certain algebraic structure (mathematics) algebraic structure - Any formal mathematical system consisting of a set of objects and operations on those objects. Examples are Boolean algebra, numerical algebra, set algebra and matrix algebra.  related to a model of the q-harmonic oscillator. We present here only the summary of results; a paper with detailed proofs will appear elsewhere [53].

To be more specific, let us consider a q-quadratic lattice of the form x = ([q.sup.s] + [q.sup.-s])/2 with [q.sup.s] = [e.sup.i[theta Theta

A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option.
]] and let us introduce the symmetric difference In mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets, but not in both. This operation is the set-theoretic equivalent of the exclusive disjunction (XOR operation) in Boolean logic.  operator as

[delta]f (x (s)) = f (x (s + 1/2)) - f (x (s - 1/2)).

The first order Askey-Wilson divided difference operator is given by

(1.1) [D.sub.q]f (x) := [delta]f (x)/[delta]x

= f (x (s + 1/2)) - f (x (s - 1/2))/x (s + 1/2) - x (s - 1/2)

Several "right" inverses [D.sup.-1.sub.q] of the Askey-Wilson divided difference operator, such that [D.sup.-1.sub.q] [D.sub.q] = I and I 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 q-extensions [8, 11, 41]. Generalizing Wiener's method to the level of 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 q-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 [sub.8][[phi].sub.7] functions.

The paper is organized as follows. In [section] 1 to [section]4 we remind the reader basic facts about the continuous q-Hermite polynomials and consider a model of q-harmonic oscillator in terms of these polynomials. In [section]5 to [section]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 of the q-Fourier transform is briefly discussed in [section]8. An explicit realization of the number operator in this model of the q-oscillator terms of Hadamard's principal values integral is outlined in [section]9. Inverses of the first order Askey-Wilson operators are constructed in [section] 10. In conclusion, the resolvent and Green's function of the corresponding q-Hamiltonian are found in [section] 11. More details can be found in the forthcoming paper [53].

2. Continuous q-Hermite Polynomials. Although the continuous q-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) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .]

and the continuous orthogonality relation is [2], [9], [10]

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

(2.3) [[integral].sup.1.sub.-1] [H.sub.n](x|q) [H.sub.m](x|q) [rho](x) dx = [d.sup.2.sub.n] [[delta].sub.mn],

where the weight function is

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the [L.sup.2]-norm is given by

(2.5) [d.sup.2.sub.n] = 2[pi] (q; q)n/(q; q) [infinity].

The continuous q-Hermite polynomials (2.1) obey a very important property, namely, the action of the Askey-Wilson divided difference operator (1.1) on [H.sub.n] (x|q) results in the same polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a  of the lower degree,

[delta]/[delta]x [H.sub.n](x|q) = [2q.sup.(1-n)/2] 1 - [q.sup.n]/1 - q [H.sub.n-1](x|q),

which is a q-analog of the familiar formula [H'.sub.n] (x) = 2n[H.sub.n-1] (x). It is worth also noting that the continuous q-Hermite polynomials are the simplest special case of the fundamental Askey-Wilson polynomials [p.sub.n] (x; a, b, c, d) [14] corresponding to the zero-valued parameters, [H.sub.n](x|q) = [p.sub.n] (x; 0, 0, 0, 0). 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.

3. Bilinear Generating Functions. There are several important generating functions for the continuous q-Hermite polynomials; see, for example, [6], [30], [48], and [51]. The Poisson kernel of Rogers, or the q-Mehler formula, is one of them

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with x = cos [theta], y = cos [phi], |t| < 1 and [d.sup.2.sub.n] defined by (2.5). A related kernel is

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Both kernels (3.1) and (3.2) are special cases k = 0 and k = 1, respectively, of a more general Carlitz's formula,

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

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 q-Hermite polynomials are the moments of the distribution function of the Al-Salam and Carlitz polynomials [3].

Let us also consider another related kernel

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and introduce the generalizations of the T, L and M kernels as follows: C* tn

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

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 q-Heisenberg algebra.

4. The q-Heisenberg Algebra. Recent advances in quantum groups has led to a study of the so-called q-harmonic oscillators, originally introduced by Arik and Coon coon: see raccoon.  [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 q-oscillator is a simple quantum mechanical system described by an annihilation annihilation

In physics, a reaction in which a particle and its antiparticle (see antimatter) collide and disappear. The annihilation releases energy equal to the original mass m multiplied by the square of the speed of light c, or E = m
 operator and a creation operator parameterized by a parameter q. 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 q-oscillator in a Hilbert space Noun 1. Hilbert space - a metric space that is linear and complete and (usually) infinite-dimensional
metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the
 of analytic functions was discussed in [7]. Later introduced models of q-oscillators are closely related to the q-orthogonal polynomials. The q-analogs of boson boson: see elementary particles; Bose-Einstein statistics.
boson

Subatomic particle with integral spin that is governed by Bose-Einstein statistics.
 operators have been studied by various authors and the corresponding wave functions were constructed in terms of the continuous q-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 q-Charlier polynomials of Al-Salam and Carlitz [3] by Askey and Suslov [12], [13] and by Zhedanov [61]. The model related to the Rogers-Szeg6 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 q-Hermite polynomials where the weight function [rho](x) given by (2.4) is continuous and positive on (-1,1) and the corresponding wave functions form a complete system.

Just as the Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, as the eigenstates of the quantum  [H.sub.n](x) are associated with the wave functions for the harmonic oscillator Harmonic oscillator

Any physical system that is bound to a position of stable equilibrium by a restoring force or torque proportional to the linear or angular displacement from this position.
 [36], the continuous q-Hermite polynomials [H.sub.n](x|q) are associated with the normalized q-wave function for the q-harmonic oscillator,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that the orthogonality relation (2.2)-(2.5) now reads

[[integral].sup.1.sub.-1] [[PSI].sub.n] (x|q) [[PSI].sub.n](x|q) dx = [[delta].sub.mn].

The q-annihilation operator [a.sub.q] (x) and the q-creation operator [a.sup.+.sub.q] (x) that satisfy the commutation rule

[a.sub.q](x)[a.sup.+.sub.q] (x) - [q.sup.-1] [a.sup.+.sub.q] (x)[a.sub.q](x) = 1

were introduced explicitly in [ 15]. In this paper we shall consider another form of the q-boson operators which is equivalent to those given in [15], but more convenient for our purposes; see [28].

The q-annihilation operator a = [a.sub.q] (x) and the q-creation operator b = [a.sup.+.sub.q] (x) satisfy the commutation rule

(4.1) ab - [q.sup.-1]ba = 1

and act on the corresponding q-wave functions as follows

(4.2) a|n) = [(1 - [q.sup.-n]/1 - [q.sup.-1]).sup.1/2] |n - 1),

(4.3) b|n) = [(1 - [q.sup.-n-1]/1 - [q.sup.-1]).sup.1/2] |n + 1).

Let S be a space of analytic functions spanned by {[H.sub.n] (x|q)}|[sup.[infinity][sub.n=0] and let the weighted inner product in S be

(4.4) [([psi], [chi]).sub.p] := [[integral].sup.1.sub.-1] [psi]* (x) [chi] (x) [rho](x) dx,

where * denotes the complex conjugate complex conjugate
n.
Either one of a pair of complex numbers whose real parts are identical and whose imaginary parts differ only in sign; for example, 6 + 4i and 6 - 4i are complex conjugates.

Noun 1.
; we need also impose certain analyticity condition on the functions [psi] and [chi] [53]; see also [52], [45] for the maximum domain of analyticity of the series in the continuous q-Hermite polynomials.

In this paper we consider the following explicit realization of the q-annihilation a and q-creation b operators. One can easily verify that the divided difference operators

(4.5) a = - [q.sup.1/2]/[(1 - q).sup.1/2] [([q.sup.s] - [q.sup.-s]).sup.-1] ([e.sup.1/2[delta]] - [e.sup.1/2[delta]])

(4.6) b = - 1/[(1 - q).sup.1/2] [([q.sup.s] - [q.sup.-s]).sup.-1] ([q.sup.-2s] [e.sup.1/2[delta]] - [q.sup.-2s][e-.sup.1/2[delta]]),

acting on analytic functions of the form

[psi](s) = [PSI](x(s)), x(s) = ([q.sup.s] + [q.sup.-s])/2,

where exp exp
abbr.
1. exponent

2. exponential
 ([alpha] [delta]) is the shift operator,

exp ([alpha] [delta]) [psi](s) = [psi](s + [alpha]),

indeed, satisfy the q-commutation rule (4.1). Moreover, it is easy to see that these operators are adjoint Ad´joint

n. 1. An adjunct; a helper.
 to each other,

[(b[psi], [chi]).sub.[rho]] = [([psi], [a.sub.[chi]]).sub.[rho]],

with respect to the inner product (4.4) in the space of analytic functions under consideration.

5. Introducing Integral Operators. Using the T, L and M kernels given by (3.1)(3.2), (3.4) for |t| < 1, let us consider the following integral operators

(5.1) T(t) [psi](x) = [[integral].sup.1.sub.-1] [T.sub.t] (x, y) [psi](y) [rho](y) dy,

(5.2) A(t) [psi](x) = [[integral].sup.1.sub.-1] [L.sub.t] (x, y) [psi](y) [rho](y) dy,

(5.3) C(t) [psi](x) = [[integral].sup.1.sub.-1] [M.sub.t] (y, x) [psi](y) [rho](y) dy.

and the corresponding adjoint operators

(5.4) B(t) [psi](x) = [[integral].sup.1.sub.-1] [L.sub.t] (y, x) [psi](y) [rho](y) dy,

(5.5) D(t) [psi](x) = [[integral].sup.1.sub.-1] [M.sub.t] (x, y) [psi](y) [rho](y) dy.

with respect to the inner product (4.4). Indeed,

[(A[psi], [chi]).sub.[rho]] = [([psi], B[chi]).sub.[rho]], [(C[psi], [chi]).sub.[rho]] = [([psi], D[chi]).sub.[rho]]

by the Fubini theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.  when |t| < 1; see, for example, [1], [23], [35], [37], [56] for an extensive theory of the integral operators.

In a more general setting, let us introduce also

[A.sup.(k)] [psi](x) = [[integral].sup.1.sub.-1] [L.sup.(k).sub.t] (x, y) [psi](y) [rho](y) dy,

[B.sup.(k)] [psi](x) = [[integral].sup.1.sub.-1] [L.sup.(k).sub.t] (x, y) [psi](y) [rho](y) dy,

[C.sup.(k)] [psi](x) = [[integral].sup.1.sub.-1] [M.sup.(k).sub.t] (x, y) [psi](y) [rho](y) dy,

[D.sup.(k)] [psi](x) = [[integral].sup.1.sub.-1] [M.sup.(k).sub.t] (x, y) [psi](y) [rho](y) dy,

and with the help of (3.6) verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Once again,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in the space of analytic functions, if |t| < 1.

It is easy to show that

(5.6) [A.sup.(k)] (t)[H.sub.m](x|q) = [t.sup.m-k] [(q; q).sub.m]/[(q; q).sub.m-k] [H.sub.m-k] (x|q),

(5.7) [B.sup.(k)] (t)[H.sub.m](x|q) = [t.sup.m] [t.sub.m+k](x|q),

(5.8) [C.sup.(k)] (t)[H.sub.m](x|q) = [t.sup.m] [(q; q).sub.m]/[(q; q).sub.m+k] [H.sub.m+k] (x|q),

(5.9) [D.sup.(k)] (t)[H.sub.m](x|q) = [t.sup.m-k] [H.sub.m-k](x|q), m [greater than or equal to] k.

For k = 1 these relations (5.6)-(5.9) define a set of integral operators that correspond to the so-called raising and lowering operators for the continuous q-Hermite polynomials:

(5.10) A(t)[H.sub.m](x|q) = [t.sup.m-1] (1 - [q.sup.m]) [H.sub.m-1] (x|q),

(5.11) B(t)[H.sub.m](x|q) = [t.sup.m] [H.sub.m+1] (x|q),

(5.12) C(t)[H.sub.m](x|q) = [t.sup.m]/1 - [q.sup.m+1] [H.sub.m+1] (x|q),

(5.13) D(t)[H.sub.m](x|q) = [t.sup.m-1][H.sub.m-1](x|q), m [not equal to] 0.

The continuous q-Hermite polynomials are eigenfunctions of the T-operator:

(5.14) T(t)[H.sub.m](x|q) = [t.sup.m] [H.sub.m](x|q).

Combining (5.10) and (5.11) we find

(5.15) B([t.sub.1]) A ([t.sub.2]) [H.sub.m](x|q) = [([t.sub.1][t.sub.2]).sup.m-1] (1 - [q.sup.m]) [H.sub.m](x|q).

This integral equation with two free parameters [t.sub.1] and [t.sub.2] extends the corresponding second order difference equation for the continuous q-Hermite polynomials; see [14], [18] and [40] for more details on this equation. Another integral equation follows from (5.12)-(5.13).

6. "Algebra" of Integral Operators. The integral operators T, A, B, C, and D obey the following multiplication rules:
                T ([t.sub.2])   A ([t.sub.2])   B ([t.sub.2])

T ([t.sub.1])   Eq. (6.1)       Eq. (6.3)       Eq. (6.5)
A ([t.sub.1])   Eq. (6.2)       Eq. (6.16)      Eq. (6.12)
B ([t.sub.1])   Eq. (6.4)       Eq. (6.13)      Eq. (6.17)
C ([t.sub.1])   Eq. (6.6)       Eq. (6.11)      Eq. (6.21)
D ([t.sub.1])   Eq. (6.8)       Eq. (6.19)      Eq. (6.10)

                C ([t.sub.2])   D ([t.sub.2])

T ([t.sub.1])   Eq. (6.7)       Eq. (6.9)
A ([t.sub.1])   Eq. (6.10)      Eq. (6.18)
B ([t.sub.1])   Eq. (6.20)      Eq. (6.11)
C ([t.sub.1])   Eq. (6.22)      Eq. (6.14)
D ([t.sub.1])   Eq. (6.15)      Eq. (6.23)


All the products in this table can be evaluated directly from the definitions of the integral operators and corresponding kernels in the following manner:

(6.1) T ([t.sub.1]) T ([t.sub.2]) = T ([t.sub.1][t.sub.2]),

(6.2) A ([t.sub.1]) T ([t.sub.2]) = [t.sub.2] A ([t.sub.1][t.sub.2]),

(6.3) T ([t.sub.1]) A ([t.sub.2]) = A ([t.sub.1][t.sub.2]),

(6.4) B ([t.sub.1]) T ([t.sub.2]) = B ([t.sub.1][t.sub.2]),

(6.5) T ([t.sub.1]) B ([t.sub.2]) = [t.sub.1] B ([t.sub.1][t.sub.2]),

(6.6) C ([t.sub.1]) T ([t.sub.2]) = C ([t.sub.1][t.sub.2]),

(6.7) T ([t.sub.1]) C ([t.sub.2]) = [t.sub.1] C ([t.sub.1][t.sub.2]),

(6.8) D ([t.sub.1]) T ([t.sub.2]) = [t.sub.2] D ([t.sub.1][t.sub.2]),

(6.9) T ([t.sub.1]) D ([t.sub.2]) = D ([t.sub.1][t.sub.2]),

(6.10) A ([t.sub.1]) C ([t.sub.2]) = D ([t.sub.1]) B ([t.sub.2]) = T ([t.sub.1][t.sub.2]),

(6.11) C ([t.sub.1]) A ([t.sub.2]) = B ([t.sub.1]) D ([t.sub.2]) = [([t.sub.1][t.sub.2]).sup.-1] (T ([t.sub.1][t.sub.2]) - T(0)),

(6.12) A ([t.sub.1]) B ([t.sub.2]) = T ([t.sub.1][t.sub.2]) - qT ([t.sub.1][t.sub.2]),

(6.13) B ([t.sub.1]) A ([t.sub.2]) = [([t.sub.1][t.sub.2]).sup.-1] (T ([t.sub.1][t.sub.2]) - T ([t.sub.1][t.sub.2])),

(6.14) C ([t.sub.1]) D ([t.sub.2]) = [([t.sub.1][t.sub.2]).sup.-1] ([[infinity].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over (k=0)] T ([t.sub.1][t.sub.2][q.sup.k]) - T(0)),

(6.15) D ([t.sub.1]) C ([t.sub.2]) = [[infinity].summation over (k=0)] T ([t.sub.1][t.sub.2][q.sup.k]) [q.sup.k],

(6.16) A ([t.sub.1]) A ([t.sub.2]) = [t.sub.2] [A.sup.(2)] ([t.sub.1][t.sub.2]),

(6.17) B ([t.sub.1]) B ([t.sub.2]) = [t.sub.1] [B.sup.(2)] ([t.sub.1][t.sub.2]),

(6.18) A ([t.sub.1]) D ([t.sub.2]) = [t.sub.2] [[infinity].summation over (k=0)] [q.sup.2k] [A.sup.(2)] ([t.sub.1][t.sub.2][q.sup.k]),

(6.19) D ([t.sub.1]) A ([t.sub.2]) = [t.sub.2] [[infinity].summation over (k=0)] [q.sup.k] A ([t.sub.1][t.sub.2][q.sup.k]),

(6.20) B ([t.sub.1]) C ([t.sub.2]) = [t.sub.1] [[infinity].summation over (k=0)] [q.sup.k] [B.sup.(2)] ([t.sub.1][t.sub.2][q.sup.k]),

(6.21) C ([t.sub.1]) B ([t.sub.2]) = [t.sub.1] [[infinity].summation over (k=0)] [q.sup.2k] [B.sup.(2)] ([t.sub.1][t.sub.2][q.sup.k]),

(6.22) C ([t.sub.1]) C ([t.sub.2]) = [t.sub.1] [[infinity].summation over (k=0)] [q.sup.k] 1 - [q.sup.k+1]/1 - q [B.sup.(2)] ([t.sub.1][t.sub.2][q.sup.k]),

(6.23) D ([t.sub.1]) D ([t.sub.2]) = [t.sub.1] [[infinity].summation over (k=0)] [q.sup.k] 1 - [q.sup.k+1]/1 - q [A.sup.(2)] ([t.sub.1][t.sub.2][q.sup.k])

and so on. Here max (|[t.sub.1]|, |[t.sub.2]|) < 1, when all integral operators are bounded.

Although this "algebra" of integral operators is not closed, it unifies many important 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 q-Fourier transform [8] as special cases after certain analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series  with respect to the free parameter The introduction to this article provides insufficient context for those unfamiliar with the subject matter.
Please help [ improve the introduction] to meet Wikipedia's layout standards. You can discuss the issue on the talk page.
. 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 |t| < 1, when they are bounded. In this section we shall consider analytic continuation of these integral operators outside the interval 0 [less than equal to] t < 1. This leads to several important (unbounded) operators, when t = 1, [q.sup.-1/2], [q.sup.-1], etc.

7.1. Operator T (1). It can be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or T (1) is the identity operator

T(1) = I

in the space of analytic functions under consideration; see [53] for more details.

7.2. Operator T ([q.sup.-1/2]). In a similar fashion

(7.1) T ([q.sup.1/2]) = [q.sup.s][e.sup.-1/2 [delta]] - [q.sup.-s][e.sup.1/2 [delta]]/[q.sup.s] - [q.sup.-s].

This can be shows as a result of "collision" of the poles in the complex plane [53]. Now operators T ([q.sup.-k/2]) can be found as products of T ([q.sup.-1/2]) from (7.1):

(T ([q.sup.-1/2]))[sup.k] T ([q.sup.-k/2]).

For example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where I is the identity operator.

The operator T ([q.sup.-1]) is closely related to the Hamiltonian of the model of the q-harmonic oscillator under consideration, namely,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here a and b are the q-annihilation and q-creation operators, given by (4.5) and (4.6), respectively.

7.3. Operators A ([q.sup.-1/2] )and B ([q.sup.-1/2]). "Colliding" the poles in the complex plane [53], one can show that

A ([q.sup.-1/2]) = [(1 - q).sup.1/2] a,

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,

B ([q.sup.-1/2]) = [(1 - q).sup.1/2] b.

From the multiplication table of the integral operators we obtain the following (q)-commutators:

A ([t.sub.1]) B ([t.sub.2]) - [t.sub.1][t.sub.2]B ([t.sub.2]) A ([t.sub.1]) = (1 - q) T (q[t.sub.1][t.sub.2]),

A ([t.sub.1]) B ([t.sub.2]) - [t.sub.1][t.sub.2]B ([t.sub.2]) A ([t.sub.1]) = (1 - q) T ([t.sub.1][t.sub.2]).

The special cases [t.sub.1] = [t.sub.2] = [q.sup.-1/2] are well-known in the theory of q-oscillators [19], [38]:

ab - [q.sup.-1] ba = I, ab - ba = T ([q.sup.-1]).

Also, from the multiplication table of the integral operators,

A ([t.sub.1]) T ([t.sub.2]) = [t.sub.2]T ([t.sub.2]) A ([t.sub.1]),

B ([t.sub.1]) T ([t.sub.2]) = [t.sub.2] 1T ([t.sub.2]) B ([t.sub.1]),

and when [t.sub.1] = [q.sup.-1/2], [t.sub.2] = t one gets

aT (t) = tT (t) a,

bT (t) = [t.sup.-1]T (t) b.

In the case t - [q.sup.-1/2] we can use these relations in order to determine the spectrum of the q-Hamiltonian H in a pure algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind.

[CACM 2(5):16 (May 1959)].
2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements.
 form.

The normalized q-wave functions in the model of q-oscillator under consideration are

[[psi].sub.n] (x) = [([q.sup.n+1]; q)[sub.[infinity]]/2[pi]][sup.1/2] [H.sub.n] (x|q)

with the orthogonality relation

[[integral].sup.1.sub.-1] [[psi].sub.n] (x) [[psi].sub.m] (x) P (x) dx = [[delta].sub.mn]

and the explicit action of the q-annihilation and q-creation operators on these wave functions is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

according to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 the general rule (4.2)-(4.3).

8. Generalized q-Fourier Transform and its Inverse. The semi-group property from the multiplication table is

T ([t.sub.1]) T ([t.sub.2]) = T ([t.sub.1][t.sub.2]),

when max (|t.sub.1|, |[t.sub.2]|) < 1. "Analytic continuation" of the integral operators T ([t.sub.1,2]) on the unit circle |[t.sub.1,2]| = 1 results in the q-Fourier transform [8], [11], [41], [53] (usually, in the classical case, [tau] = [pi]/2 [55], [59], but we discuss the general case with 0 < [tau] < [pi]). Then, formally,

T ([e.sup.i[tau]]) T ([e.sup.-i[tau]]) = h

where I 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 q-Fourier transform and its inverse are certain singular integral equations, somewhat similar to the case of the classical Hilbert transform In mathematics and in signal processing, the Hilbert transform of a real-valued function, is another real-valued function in the same domain. ; see [24], [39] for an account of the theory of singular integral equations.

9. The "Number" Operator. The concept of number operator is well-known in quantum mechanics quantum mechanics: see quantum theory.
quantum mechanics

Branch of mathematical physics that deals with atomic and subatomic systems. It is concerned with phenomena that are so small-scale that they cannot be described in classical terms, and it is
 [36]. Similar operators were formally introduced in the theory of q-harmonic operators [19], [38], but explicit realizations of these "number" operators were not con structed. In the model of the q-oscillator under consideration it is natural to introduce this operator as the generator of the semi-group of the integral operators [T.sup.(t)] [53]. Denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or in the form of a contour integral,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where r is a contour corresponding to analytic continuation of the operator [T.sup.(t)] to the values |t| > 1; see [53] for the details. Then the semi-group properties can be written as usual

[T.sub.0] = I, [T.sub.[alpha]][T.sub.[beta]] = [T.sub.[alpha] + [beta]]

and formally

[T.sub.[alpha]] = exp ([alpha]I) = [[infinity].summation over (n=0)] [([alpha]I).sup.n]/n!,

where by the definition the "infinitesimal" operator is

I := ([dT.sub.[alpha]]/[d.aub.[alpha]])|[sub.[alpha]=0.

Explicit realization of the number operator I can be given in terms of Hadamard's principal value integral in the space of analytic functions; see [53] for the details.

10. Inversions of Operators A (t) and B (t). The operators A (t) and B (t) are bounded integral operators for |t| < 1; they admit an analytic continuation in the larger domain |t| > 1 and become unbounded divided difference operators when t = [q.sup.-1/2]. The problem of finding inverses of these operators is similar to the familiar classical results

d/dx [integral] f (x) dx = f (x),

[integral] d/dx f (x) dx = f (x) + constant.

In the model of the q-harmonic oscillator under consideration we can extend these relations to q-derivatives, or even to q-"fractional" derivatives, namely, our integral operators A (t) and B (t), with the help of the integral operators C (t) and D (t). Indeed, for |t| < 1 one can write from the table of multiplication of the operators that

A ([t.sup.-1]) C (t) = T (1) = I,

D (t) B ([t.sup.-1]) = T (1) = I,

where I is the identity operator. Thus the bounded integral operator C (t) (D (t)) with |t| < 1 gives the right (left) inverse of the unbounded operator A ([t.sup.-1]) (B ([t.sup.-1])), provided that this operator is properly analytically continued to the domain |[t.sup.-1]| > 1. When t [right arrow] [q.sup.-1/2] one gets, as a special case, the right inverse of the Askey-Wilson first order divided difference operator [delta]/[delta]x originally found by Brown and Ismail [20] in this model of q-oscillator. In a similar manner,

C (t) A ([t.sup.-1]) = T (1) - T (0), B ([t.sup.-1]) D (t) = T (1) - T (0),

when |t| < 1 and

A ([t.sub.1]) C ([t.sub.2]) - [t.sub.1][t.sub.2]C ([t.sub.2]) A ([t.sub.1]) = T (0),

D ([t.sub.2]) B ([t.sub.1]) - [t.sub.1][t.sub.2]B ([t.sub.1]) D ([t.sub.2]) = T (0),

i.e., these "commutators" act on a vector 0 as the projection operator to the "vacuum" vector [[psi].sub.0].Indeed,

T (0) [psi] (x) = [[integra].sup.1.sub.-1] [T.sub.0] (x, y) [psi] (y) [rho] (y) dy

= ([[psi].sub.0], [psi])[rho], [[psi].sub.0],

where [[psi].sub.0] = [d.sup.-1.sub.0] [H.sub.0] (x|q) is the "vacuum" vector.

11. Resolvents and Green's Functions. The continuous q-Hermite polynomials, or the wave functions in the model of the q-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 q-Hermite polynomials

T ([q.sup.-1/2]) [H.sub.n] (x|q) = [q.sup.-n/2][H.sub.n] (x|q),

which is the special case t = [q.sup.-1/2] of (5.14) due to (7.1), and consider

(11.1) (T([q.sup.-1/2]) - [LAMBDA The Greek letter "L," which is used as a symbol for "wavelength." A lambda is a particular frequency of light, and the term is widely used in optical networking. Sending "multiple lambdas" down a fiber is the same as sending "multiple frequencies" or "multiple colors. ]I) [psi] = [chi]

with the resolvent

[R.sub.[LAMBDA]] = [(T ([q.sup.-1/2]) - [LAMBDA]I).sup.-1], [psi] = [R.sub.[LAMBDA]] [chi].

Multiplying (11.1) by the corresponding bounded integral operator T ([q.sup.1/2]), one gets

(I - [LAMBDA]T ([q.sup.1/2])) T ([q.sup.1/2]) [chi],

where I is the identity operator. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by (6.1), and

[R.sub.[LAMBDA]] = [(T([q.sup.-1/2]) - [LAMBDA]I).sup.-1] = [[infinity].summation over (k=0)] ([q.sup.(k+1)/2]).

So, the resolvent is an integral operator

[R.sub.[LAMBDA]] [chi] (x) = [[integral].sup.1.sub.-1] [R.sub.[LAMBDA]] (x, y) [chi] (y) n (y) dy

with the kernel

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Introducing the eigenvalues eigenvalues

statistical term meaning latent root.
 An = q -n/2 and the orthonormal eigenfunctions [[psi].sub.n] = [d.sup.-1.sub.n][H.sub.n] (x|q), one can finally write

[R.sub.[LAMBDA]] (x, y) = [[infinity].summation over (n=0)] [[psi].sub.n] (x) [[psi].sub.n] (y)/[[LAMBDA].sub.n] - [LAMBDA].

The resolvent identity holds

(11.2) ([mu] - A) [R.sub.[LAMBDA]][R.sub.[mu]] = [R.sub.[LAMBDA]] - [R.sub.[mu]],

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 q-Hamiltonian) in the following manner. At the level of the integral operators in Eq. (5.15) we have

B ([t.sub.1]) A ([t.sub.2]) = [([t.sub.1][t.sub.2]).sup.-1] (T ([t.sub.1][t.sub.2]) - T (q[t.sub.1][t.sub.2]))

and, hence, for [t.sub.1] = [t.sub.2] = [q.sup.-1/2],

[q.sup.-1] B ([q.sup.-1/2]) A ([q.sup.-1/2]) = T ([q.sup.-1]) - I.

Therefore, instead of solving

([q.sup.-1]B ([q.sup.-1/2]) A ([q.sup.-1/2]) + [LAMBDA]) [psi] = [chi],

one can solve a simpler equation

(11.3) (T ([q.sup.-1]) - [mu]I) [psi] = [chi], [mu] = 1 - [LAMBDA]

with the help of the resolvent

[R.sub.[mu]] = [(T([q.sup.-1]) - [mu]I).sup.-1], [psi] = [R.sub.[mu]] [chi].

Multiplying (11.3) by T (q),

(I - [mu]T (q)) [psi] = T (q) [chi],

where I is the identity operator, and once again

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This consideration gives an explicit representation for the resolvent in terms of the integral operator

[R.sub.[mu]] [chi] (x) = [[integral].sup.1.sub.-1] [R.sub.[mu]] (x, y) [chi] (y) [rho](y) dy

with the kernel

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The resolvent identity (11.2) holds.

11.3. Green's function. Let

[chi] = [delta](x-x'),

where S (y) is the Dirac delta function The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x . Then

[G.sub.[mu]] (x, x') = [R.sub.[mu]][delta] (x - x') = [R.sub.[mu]] (x, x') [rho] (x'),

or

(T ([q.sup.-1]) - [mu]I) [G.sub.[mu]] (x, y) = [delta] (x - y)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

See [53] for more details.

Acknowledgement. This work was completed when the author visited Department of Mathematics and Statistics at Carleton University Carleton University, at Ottawa, Ont., Canada; nonsectarian; coeducational; founded 1942 as Carleton College. It achieved university status in 1957. It has faculties of arts, social sciences, science, engineering, and graduate studies, as well as the Centre for , 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.

* Received May 30, 2003. Accepted for publication January 10, 2004. Recommended by F. Marcelldn.

REFERENCES

[1] N. I. AKHIEZER AND I. M. GLAZMAN, Theory of Linear Operators in Hilbert Space, Dover, New York Dover is a town in Dutchess County, New York, United States. The population was 8,565 at the 2000 census. The town was named after Dover in England, the home town of an early settler.

The Town of Dover is located on the eastern boundary of the county.
, 1993.

[2] W. R. ALLAWAY, some properties of the q-Hermite polynomials, Canad. J. Math., 32 (1980), pp. 686-694.

[3] W. A. AL-SALAM AND L. CARLITZ, Some orthogonal q-polynomials, Math. Nachr., 30 (1965), pp. 47-61.

[4] W. A. AL-SALAM AND T. S. CHIHARA, Convolutions of orthogonal polynomials, SIAM J. Math. Anal.,, 7 (1976), pp. 16-28.

[5] W. A. AL-SALAM AND M. E. H. ISMAIL, q-Beta integrals and the q-Hermite polynomials, Pacific J. Math., 135 (1988), pp. 209-221.

[6] G. E. ANDREWS, R. A. ASKEY, AND R. ROY, Special Functions In mathematics, special functions are particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. , Cambridge University Press Cambridge University Press (known colloquially as CUP) is a publisher given a Royal Charter by Henry VIII in 1534, and one of the two privileged presses (the other being Oxford University Press). , Cambridge, 1999.

[7] M. ARIK AND D. D. COON, Hilbert spaces of analytic functions and generalized coherent states, J. Math, Phys., 17 (1976), pp. 524-527.

[8] R. A. AS KEY, N. M. ATAKISHIYEV, AND S. K. SUSLOV, An analog of the Fourier transformation for the q-harmonic oscillator, in Symmetries in Science, VI, Plenum, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
, 1993, pp. 57-63.

[9] R. A. ASKEY AND M. E. H. ISMAIL, A generalization of ultraspherical polynomials, in Studies in Pure Mathematics, P. Erdds, ed., Birkhduser, Boston, Mass., 1983, pp. 55-78.

[10] R. A. AS KEY AND M. E. H. ISMAIL, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc., 49 (1984), pp. iv+108.

[11] R. A. AS KEY, M. RAHMAN, AND S. K. SUSLOV, On a general q-Fouriertransformation with nonsymmetric kernels, J. Comp. Appl. Math., 68 (1996), pp. 25-55.

[12] R. A. AS KEY AND S. K. SUSLOV, The q-harmonic oscillator and an analogue of the Charlier polynomials, J. Phys. A., 26 (1993), pp. L693-L698.

[13] R. A. ASKEY AND S. K. SUSLOV, The q-harmonic oscillator and the Al-Salam and Carlitz polynomials, Lett. Math. Phys., 29 (1993), pp. 123-132.

[14] R. A. ASKEY AND J. A. WILSON, Some basic hypergeometric orthogonal polynomials that generalize generalize /gen·er·al·ize/ (-iz)
1. to spread throughout the body, as when local disease becomes systemic.

2. to form a general principle; to reason inductively.
 Jacobi polynomials In mathematics, Jacobi polynomials are a class of orthogonal polynomials. They are obtained from hypergeometric series in cases where the series is in fact finite:

, Mem. Amer. Math. Soc., 54 (1985), pp. iv+55.

[15] N. M. ATATISHIYEV AND S. K. SUSLOV, Difference analogs of the harmonic oscillator, Theoret. and Math. Phys., 85 (1990), pp. 1055-1062.

[16] N. M. ATAKISHIYEV AND S. K. SUSLOV, Difference hypergeometric functions, in Progress in Approximation Theory In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterising the errors introduced thereby. Note that what is meant by best and simpler will depend on the application. , Springer Series in Computational Mathematics Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. , 19, Springer-Verlag, New York, 1992, pp. 1-35.

[17] N. M. ATATISHIYEV AND S. K. SUSLOV, A realization of the q-harmonic oscillator, Theoret. and Math. Phys., 87 (1990), pp. 442-444.

[18] N. M. ATAKISHIYEV AND S. K. SUSLOV, On the Askey-Wilson polynomials, Constr. Approx., 8 (1992), pp. 363-369.

[19] L. C. BIEDENHARN, The quantum group In mathematics and theoretical physics, quantum groups are certain noncommutative algebras that first appeared in the theory of quantum integrable systems, and which were then formalized by Vladimir Drinfel'd and Michio Jimbo.  SUQ suq  
n.
Variant of souk.
 (2) and a q-analogue of the boson operators, J. Phys. Math. A., 123 (1989), pp. L873-L878.

[20] B. M. BROWN AND M. E. H. ISMAIL,A right inverse of the Askey-Wilson operator, Proc. Amer. Math. Soc., 123 (1995), pp. 2071-2079.

[21] J. BUSTOZ AND S. K. SUSLOV, Basic analog of Fourier series Fourier series

In mathematics, an infinite series used to solve special types of differential equations. It consists of an infinite sum of sines and cosines, and because it is periodic (i.e.
 on a q-quadratic grid, Methods Appl. Anal., 5 (1998), pp. 1-38.

[22] L. CARLITZ, Generating functions for certain q-orthogonal polynomials, Collect. Math., 23 (1972), pp. 91104.

[23] R. COURANT Cou`rant´   

a. 1. (Her.) Represented as running; - said of a beast borne in a coat of arms.
n. 1. A piece of music in triple time; also, a lively dance; a coranto.
2.
 AND D. HILBERT, Methods of Mathematical Physics, John Wiley John Wiley may refer to:
  • John Wiley & Sons, publishing company
  • John C. Wiley, American ambassador
  • John D. Wiley, Chancellor of the University of Wisconsin-Madison
  • John M. Wiley (1846–1912), U.S.
, New York, 1937.

[24] R. ESTRADA AND R. P. KANWAL, Singular Integral Equations, Birkhduser, Boston, 2000.

[25] R. FLOREANINI, J. LETOURNEUX, AND L. VINET, An algebraic interpretation of the continuous big qHermite polynomials, J. Math. Phys., 36 (1995), pp. 5091-5097.

[26] R. FLOREANINI, J. LETOURNEUX, AND L. VINET, More on the q-oscillator algebra and q-orthogonal polynomials, J. Phys. A., 28 (1995), pp. L287-L293.

[27] R. FLOREANINI AND L. VINET, q-Orthogonal polynomials and the oscillators quantum group, Lett. Math. Phys., 22 (1991), pp. 45-54.

[28] R. FLOREANINI AND L. VINET, Automorphisms of the q-oscillator algebra and basic orthogonal polynomials, Phys. Lett. A,180 (1993), pp. 393-401.

[29] R. FLOREANINI AND L. VINET, A model for the continuous q-ultraspherical polynomials, J. Math. Phys., 36 (1995), pp. 3800-3813.

[30] G. GASPER gasp·er  
n. Chiefly British Slang
A cigarette.
 AND M. RAHMAN, Basic Hypergeometric Series In mathematics, the basic hypergeometric series, also sometimes called the hypergeometric q-series, are q-analog generalizations of ordinary hypergeometric series. Two basic series are commonly defined, the unilateral basic hypergeometric series, and the , Cambridge University Press, Cambridge, 1990.

[31] M. E. H. ISMAIL, M. RAHMAN, AND R. ZHANG, Diagonalization of certain integral operators II, J. Comp. Appl. Math., 68 (1996), pp. 163-196.

[32] M. E. H. ISMAIL, D. STANTON, AND G. VIENNOT, The combinatorics combinatorics (kŏm'bənətôr`ĭks) or combinatorial analysis (kŏm'bĭnətôr`ēəl)  of the Askey-Wilson integral, J. Comp. Appl. Math., 68 (1996), pp. 163-196.

[33] M. E. H. ISMAIL AND R. ZHANG, Diagonalization of certain integral operators, Adv. Math., 108 (1994), pp. 1-33.

[34] R. KOEKOEK AND R. F. SWARTTOW, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogues, Report 98-17, Delft University of Technology Delft University of Technology, (Technische Universiteit Delft in Dutch) in Delft, the Netherlands, is the largest and most comprehensive technical university in the Netherlands, with over 13,000 students and 2,100 scientists (including 200 professors). , 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. KOLMOGOROV AND S. V. FOMIN, Introductory Real Analysis, Dover, New York, 1970.

[36] L. D. LANDAU lan·dau  
n.
1. A four-wheeled carriage with front and back passenger seats that face each other and a roof in two sections that can be lowered or detached.

2. A style of automobile with a similar roof.
 AND E. M. LIFSCHITZ, Quantum Mechanics: Non-relativistic Theory, Addison-Wesley, 1998.

[37] P. D. LAx, Functional Analysis, John Wiley, New York, 2002.

[38] A. J. MACFARLANE MacFarlane or Macfarlane is a surname shared by:
  • Alan Macfarlane (born 1941), a professor of anthropological science at Cambridge University
  • Alexander Macfarlane (mathematician) (1851-1913), a Scottish-Canadian logician, physicist, and mathematician
, On q-analogues of the quantum harmonic oscillator The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable  and the quantum group SU(2)Q, J. Phys. A., 22 (1989), pp. 4581-4588.

[39] N. I. MUSKHELISHVILI,Singular Integral Equations, Groningen-Holland, 1953.

[40] A. F. NIKIFOROV, S. K. SUSLOV, AND V. B. UVARGV, Classical Orthogonal Polynomials of a Discrete Variable Discrete variable

Variable like 1, 2, 3. Bond ratings are examples of discrete classifications.
, Springer-Verlag, Berlin, 1991.

[41] M. RAHMAN AND S. K. SUSLOV, Singular analogue of the Fourier transformation for the Askey-Wilson polynomials, in Symmetries and Integrability of Difference Equations, CRM (Customer Relationship Management) An integrated information system that is used to plan, schedule and control the presales and postsales activities in an organization.  Proceedings and Lecture Notes, 9, Amer. Math. Soc., Providence, RI, 1996, pp. 289-302.

[42] L. J. RGGERS, Second memoir on the expansion of certain infinite products, in Proc. London Math. Soc., 25 (1894), pp. 318-343.

[43] L. J. RGGERS, Third memoir on the expansion of certain infinite products, in Proc. London Math. Soc., 26 (1895),pp.15-32.

[44] L. J. RGGERS, On two theorems of combinatory analysis and some allied identities, in Proc. London Math. Soc., 16 (1917), pp. 315-336.

[45] M. SIMON Simon, in the Bible.

1 One of the Maccabees.

2 or Simon Peter: see Peter, Saint.

3 See Simon, Saint.

4 Kinsman of Jesus.

5 Leper of Bethany in whose house a woman anointed Jesus' feet.
 AND S. K. SUSLOV, Expansion of analytic functions in q-orthogonal polynomials, J. Funct. Anal. Approx. Theory, to appear.

[46] T. J. STIELTJES, Recherches Sur Les Fractions Continues, Annales de la Faculty des Sciences de Toulouse, No. 9,1895.

[47] S. K. SUSLOV, The theory of difference analogues of special functions of hypergeometric type, Russian Math. Surveys, 44 (1989), pp. 227-278.

[48] S. K. SUSLOV, "Addition" theorems for some q-exponential and q-trigonometric functions, Methods Appl. Anal., 4 (1997), pp. 11-32.

[49] S. K. SUSLOV, Some orthogonal very-well-poised 8(P7 functions, J. Phys. A., 30 (1997), pp. 5877-5885.

[50] S. K. SUSLOV, Some orthogonal very-well-poised 8(P7 functions that generalize Askey-Wilson polynomials, Ramanujan J., 5 (2001), pp. 183-218.

[51] S. K. SUSLOV, An introduction to basic Fourier series, Developments in Mathematics, Vol. 9, Kluwer Academic, Dordrecht, 2003.

[52] S. K. SUSLOV, An analog of the Cauchy-Hadamard formula for expansions in q-polynomials, in Theory and Applications of Special Functions,A M. E. H. Ismail and Erik Koelink, eds., Developments in Mathematics, Vol. 13, Springer, New York, 2005, pp. 443-460. A

[53] S. K. SUSLOV, An algebra of integral operators in a model of q-harmonic oscillator, under preparation.

[54] G. SZEG6, Beitrag zur Theoree 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. TITCHMARCH, Introduction to the Theory ofFourier Integrals, 2nd ed., Oxford University Press, 1948.

[56] F. G. TRICOMI, Integral Equations, Dover, New York, 1985.

[57] J. VAN DER JEUGHT, The q-boson operator algebra In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space (such as a Banach space), which is typically required to be closed in a specified operator topology.  and q-Hermite polynomials, Lett. Math. Phys., 24 (1992), pp. 267-274.

[58] E. T. WHITTAKER AND G. N. WATSON Wat·son , James Dewey Born 1928.

American biologist who with Francis Crick proposed a spiral model, the double helix, for the molecular structure of DNA. He shared a 1962 Nobel Prize for advances in the study of genetics.
, A Course ofModern Analysis, Cambridge University Press, Cambridge, 1952.

[59] N. WIENER, The Fourier Integral and Certain of Its Applications, Cambridge University Press, Cambridge, 1933.

[60] S. WIGERT,A, Arkiv far Matematik, Astronomi och Fysik, 17 (1923), pp. 1-15.

[61] A. ZHEDANOV, Weyl shift of q-oscillator and q-polynomials, Theoret. and Math. Phys., 94 (1993), pp. 219224.

SERGEI K. SUSLOV ([dagger])

([dagger]) Department of Mathematics and Statistics, Arizona State University Arizona State University, at Tempe; coeducational; opened 1886 as a normal school, became 1925 Tempe State Teachers College, renamed 1945 Arizona State College at Tempe. Its present name was adopted in 1958. , 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.
COPYRIGHT 2007 Institute of Computational Mathematics
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. All rights reserved.

 Reader Opinion

Title:

Comment:



 

Article Details
Printer friendly Cite/link Email Feedback
Author:Suslov, Sergei
Publication:Electronic Transactions on Numerical Analysis
Date:Apr 1, 2007
Words:6699
Previous Article:The method of lower and upper solutions for periodic and anti-periodic difference equations.
Next Article:Periodic points of some algebraic maps.



Related Articles
Representation theory of finite groups and associative algebras. (reprint, 1962).
Operator Theory and Ill-posed Problems.
Certain number-theoretic episodes in algebra.
Crossed products of C*-algebras.
Applied vector analysis, 2d ed.
A first course in functional analysis.
Integral and functional analysis.
A polynomial collocation method for Cauchy singular integral equations over the interval.
Vertex operator algebras and related areas; proceedings.

Terms of use | Copyright © 2014 Farlex, Inc. | Feedback | For webmasters