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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].
qHermite polynomials and classical orthogonal polynomials
 Can. J. Math
, 1996
"... We use generating functions to express orthogonality relations in the form of qbeta integrals. The integrand of such a qbeta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous qHermite polynomials, the AlSala ..."
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Cited by 13 (3 self)
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We use generating functions to express orthogonality relations in the form of qbeta integrals. The integrand of such a qbeta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous qHermite polynomials, the AlSalamCarlitz polynomials, and the polynomials of Szegő and leads naturally to the AlSalamChihara polynomials then to the AskeyWilson polynomials, the big qJacobi polynomials and the biorthogonal rational functions of AlSalam and Verma, and some recent biorthogonal functions of AlSalam and Ismail.
Partialsum analogues of the RogersRamanujan identities
 J. Combin. Theory Ser. A
"... Dedicated to Barry McCoy on the occasion of his sixtieth birthday Abstract. A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the productside of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sumside by a weighted sum over Schur polynomials. ..."
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Cited by 4 (0 self)
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Dedicated to Barry McCoy on the occasion of his sixtieth birthday Abstract. A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the productside of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sumside by a weighted sum over Schur polynomials.
On the completeness of some subsystems of qdeformed coherent states, Helv
 Phys. Acta
"... The von Neumann type subsystems of qdeformed coherent states are considered. The completeness of such subsystems is proved. 1This is slightly modified version of the paper [Pe 1996] published in Helv. Phys. Acta ..."
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Cited by 2 (0 self)
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The von Neumann type subsystems of qdeformed coherent states are considered. The completeness of such subsystems is proved. 1This is slightly modified version of the paper [Pe 1996] published in Helv. Phys. Acta
qBERNOULLI AND qSTIRLING NUMBERS, AN UMBRAL APPROACH
"... The aim of this paper is to describe how different qdifference operators combine with qBernoulli, qEuler and qStirling numbers to form various qformulas. The Bernoulli numbers were first used by Jacob Bernoulli (1654 ..."
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Cited by 1 (1 self)
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The aim of this paper is to describe how different qdifference operators combine with qBernoulli, qEuler and qStirling numbers to form various qformulas. The Bernoulli numbers were first used by Jacob Bernoulli (1654
unknown title
, 2000
"... A counterintuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter. 1 A formula of Gauss revisited Consider the Newton binomial for a positive integer N: (1 − x) N = N∑ ..."
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A counterintuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter. 1 A formula of Gauss revisited Consider the Newton binomial for a positive integer N: (1 − x) N = N∑
Ladder Operators For Szegő Polynomials and
, 1994
"... We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to 4φ3 biorthogonal rational functions on the unit circle. ..."
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We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to 4φ3 biorthogonal rational functions on the unit circle.
RogersRamanujan and the . . .
, 2003
"... In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Padé approximants converges uniformly in compact subsets of the ball omitting poles of f. There is also apparently a cruder version of the conjecture due to Padé himself, goi ..."
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In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Padé approximants converges uniformly in compact subsets of the ball omitting poles of f. There is also apparently a cruder version of the conjecture due to Padé himself, going back to the early twentieth century. We show here that for carefully chosen q on the unit circle, the RogersRamanujan continued fraction 1 + qz 1 + q2z 1 + q3z 1 provides a counterexample to the conjecture. We also highlight some other interesting phenomena displayed by this fraction.
Algebraic properties of RogersSzegö functions: I. Applications in quantum optics
, 903
"... Abstract. By means of a wellestablished algebraic framework, RogersSzegö functions associated with a circular geometry in the complex plane are introduced in the context of qspecial functions, and their properties are discussed in details. The eigenfunctions related to the coherent and phase stat ..."
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Abstract. By means of a wellestablished algebraic framework, RogersSzegö functions associated with a circular geometry in the complex plane are introduced in the context of qspecial functions, and their properties are discussed in details. The eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of RogersSzegö functions, the coefficients being determined through proper eigenvalue equations in each situation. Furthermore, a complementary study on the RobertsonSchrödinger and symmetrical uncertainty relations for the cosine, sine and nondeformed number operators is also conducted, corroborating, in this way, certain features of qdeformed coherent states.
unknown title
, 2002
"... www.elsevier.com/locate/cam Mellin transforms for some families of qpolynomials ..."
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www.elsevier.com/locate/cam Mellin transforms for some families of qpolynomials