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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
On qorthogonal polynomials, dual to little and big qJacobi polynomials
 J. Math. Anal. Appl
"... This paper studies properties of qJacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra Uq(su1,1). Spectra and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthonormal basis i ..."
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Cited by 10 (3 self)
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This paper studies properties of qJacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra Uq(su1,1). Spectra and eigenfunctions of these operators are found explicitly. These eigenfunctions, when normalized, form an orthonormal basis in the representation space. The initial Uq(su1,1)basis and the bases of these eigenfunctions are interconnected by matrices whose entries are expressed in terms of little or big qJacobi polynomials. The orthogonality between lines in these unitary connection matrices leads to the orthogonality relations for little and big qJacobi polynomials. The orthogonality of rows in the connection matrices leads to an explicit form of orthogonality relations on the countable set of points for 3φ2 and 3φ1 polynomials, which are dual to big and little qJacobi polynomials, respectively. The orthogonality measure for the dual little qJacobi polynomials proves to be extremal, whereas the measure for the dual big qJacobi polynomials is not extremal. Key words. Orthogonal qpolynomials, little qJacobi polynomials, big qJacobi polynomials, Leonard pairs, orthogonality relations, quantum algebra AMS subject classification. 33D80, 33D45, 17B37 1.
Big qLaguerre and qMeixner polynomials and representations of the quantum algebra Uq(su1,1
 J. Phys. A
, 2003
"... Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra Uq(su1,1) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These eigenfunctions, when normalized, constitute an orthonormal basis in the ..."
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Cited by 6 (3 self)
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Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra Uq(su1,1) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial Uq(su1,1)basis and the basis of eigenfunctions are interrelated by a matrix with entries, expressed in terms of big qLaguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big qLaguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of qMeixner polynomials Mn(x; b, c; q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for the qMeixner polynomials Mn(x; b, c; q) with b < 0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived. PACS numbers: 02.20.Uw, 02.30.Gp, 03.65.Fd 1.
773–778, Menlo Park
 J. Math. Anal. Appl
, 1994
"... We show that a confluent case of the big qJacobi polynomials Pn(x; a, b, c; q):= 3φ2(q −n, abq n+1, x; aq, cq; q, q), which corresponds to a = b = −c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0 < a < q −1). Since Pn(x; ..."
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Cited by 2 (1 self)
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We show that a confluent case of the big qJacobi polynomials Pn(x; a, b, c; q):= 3φ2(q −n, abq n+1, x; aq, cq; q, q), which corresponds to a = b = −c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0 < a < q −1). Since Pn(x; q α, q α, −q α; q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q → 1, this family represents yet another qextension of these classical polynomials, different from the continuous qultraspherical polynomials of Rogers. The dual family with respect to the polynomials Pn(x; a, a, −a; q) (i.e., the dual discrete qultraspherical polynomials) corresponds to the indeterminate moment problem, that is, these polynomials have infinitely many orthogonality relations. We find orthogonality relations for these polynomials, which have not been considered before. In particular, extremal orthogonality measures for these polynomials are derived. 1.
Duality of qpolynomials, orthogonal on countable sets of points
, 2005
"... We review properties of qorthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (selfadjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally interpret the duality of families of polynomials, orthogonal on ..."
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Cited by 2 (0 self)
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We review properties of qorthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (selfadjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally interpret the duality of families of polynomials, orthogonal on countable sets of points. In order to obtain orthogonality relations for dual sets of polynomials, we propose to use two symmetric (selfadjoint) operators, representable (in some distinct bases) by Jacobi matrices. To illustrate applications of this approach, we apply it to several pairs of dual families of qpolynomials, orthogonal on countable sets, from the qAskey scheme. For each such pair, the corresponding operators, representable by Jacobi matrices, are explicitly given. These operators are employed in order to find explicitly sets of points, on which the polynomials are orthogonal, and orthogonality relations for them.
unknown title
, 2003
"... A set of orthogonal polynomials, dual to alternative qCharlier polynomials ..."
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A set of orthogonal polynomials, dual to alternative qCharlier polynomials
unknown title
, 2007
"... On factorization of qdifference equation for continuous qHermite polynomials ..."
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On factorization of qdifference equation for continuous qHermite polynomials
ITP979E qalg/9709035 REPRESENTATIONS OF THE qDEFORMED ALGEBRA U ′ q(so2,2)
, 1997
"... The main aim of this paper is to give classes of irreducible infinite dimensional representations and of irreducible ∗representations of the qdeformed algebra U ′ q (so2,2) which is a real form of the nonstandard deformation U ′ q (so4) of the universal enveloping algebra U(so(4,C)). These repres ..."
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The main aim of this paper is to give classes of irreducible infinite dimensional representations and of irreducible ∗representations of the qdeformed algebra U ′ q (so2,2) which is a real form of the nonstandard deformation U ′ q (so4) of the universal enveloping algebra U(so(4,C)). These representations are described by two complex parameters and are obtained by ”analytical continuation ” of the irreducible finite dimensional representations of the algebra U ′ q (so4) in the basis corresponding to the reduction from U ′ q(so4) to U(so2 ⊕ so2). 1