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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 65 (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].
The Bivariate RogersSzegö Polynomials
, 2006
"... Abstract. We obtain Mehler’s formula and the Rogers formula for the continuous big qHermite polynomials Hn(x;aq). Instead of working with the polynomials Hn(x;aq) directly, we consider the equivalent forms in terms of the bivariate RogersSzegö polynomials hn(x,yq) recently introduced by Chen, F ..."
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Cited by 2 (0 self)
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Abstract. We obtain Mehler’s formula and the Rogers formula for the continuous big qHermite polynomials Hn(x;aq). Instead of working with the polynomials Hn(x;aq) directly, we consider the equivalent forms in terms of the bivariate RogersSzegö polynomials hn(x,yq) recently introduced by Chen, Fu and Zhang. It turns out that Mehler’s formula for Hn(x;aq) involves a 3φ2 sum, and the Rogers formula involves a 2φ1 sum. The proofs of these results are based on parameter augmentation with respect to the qexponential operator and the homogeneous qshift operator in two variables. Keywords: The bivariate RogersSzegö polynomials, the continuous big qHermite polynomials, the Cauchy polynomials, the qexponential operator, the homogeneous qshift operator AMS Classification: 05A30, 33D45 1.
GENERALIZED GALOIS NUMBERS, INVERSIONS, LATTICE PATHS, FERRERS DIAGRAMS AND LIMIT THEOREMS
"... Abstract. Bliem and Kousidis recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables, using inversions in random words, random lattice paths and random Ferr ..."
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Abstract. Bliem and Kousidis recently considered a family of random variables whose distributions are given by the generalized Galois numbers (after normalization). We give probabilistic interpretations of these random variables, using inversions in random words, random lattice paths and random Ferrers diagrams, and use these to give new proofs of limit theorems as well as some further limit results. 1.
The Cauchy Operator for Basic Hypergeometric Series
, 705
"... We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears ’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can ..."
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We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears ’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bDq). Using this operator, we obtain extensions of the AskeyWilson integral, the AskeyRoy integral, Sears ’ twoterm summation formula, as well as the qanalogues of Barnes ’ lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate RogersSzegö polynomials, or the continuous big qHermite polynomials. Keywords: qdifference operator, the Cauchy operator, the AskeyWilson integral, the AskeyRoy integral, basic hypergeometric series, parameter augmentation. AMS Subject Classification: 05A30, 33D05, 33D15 1
Heine transformations for a new kind of basic hypergeometric series in U(n)
, 1994
"... Heine transformations are proved for a new kind of multivariate basic hypergeometric series which had been previously introduced by Krattenthaler in connection with generating functions for nonintersecting lattice paths. As a consequence, a qGauss and qChuVandermonde sum are proved and also a gen ..."
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Heine transformations are proved for a new kind of multivariate basic hypergeometric series which had been previously introduced by Krattenthaler in connection with generating functions for nonintersecting lattice paths. As a consequence, a qGauss and qChuVandermonde sum are proved and also a generalization of Ramanujan's 1~01 sum.