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The Askeyscheme of hypergeometric orthogonal polynomials and its qanalogue
, 1998
"... We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erent ..."
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Cited by 376 (4 self)
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We list the socalled Askeyscheme of hypergeometric orthogonal polynomials and we give a q analogue of this scheme containing basic hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation, the second order di#erential or di#erence equation, the forward and backward shift operator, the Rodriguestype formula and generating functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give the limit relations between di#erent classes of orthogonal polynomials listed in the Askeyscheme. In chapter 3 we list the qanalogues of the polynomials in the Askeyscheme. We give their definition, orthogonality relation, three term recurrence relation, second order di#erence equation, forward and backward shift operator, Rodriguestype formula and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally, in chapter 5 we...
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].
8 Lectures on Quantum Groups and qSpecial Functions
, 1996
"... The lecture notes contains an introduction to quantum groups, qspecial functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the AskeyWilson polynomials out in detail ..."
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Cited by 9 (2 self)
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The lecture notes contains an introduction to quantum groups, qspecial functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the AskeyWilson polynomials out in detail as the main example. As an application we derive an addition formula for a twoparameter subfamily of AskeyWilson polynomials. A relation between the AlSalam and Chihara polynomials and the quantised universal enveloping algebra for su(1, 1) is given. Finally, more examples and other approaches as well as some open problems are given.
Spectral analysis and the Haar functional on the quantum SU(2
 group, Commun. Math. Phys
, 1996
"... Abstract. The Haar functional on the quantum SU(2) group is the analogue of invariant integration on the group SU(2). If restricted to a subalgebra generated by a selfadjoint element the Haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a com ..."
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Cited by 6 (1 self)
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Abstract. The Haar functional on the quantum SU(2) group is the analogue of invariant integration on the group SU(2). If restricted to a subalgebra generated by a selfadjoint element the Haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both. These results by Woronowicz and Koornwinder have been proved by using the corepresentation theory of the quantum SU(2) group and Schur’s orthogonality relations for matrix elements of irreducible unitary corepresentations. These results are proved here by using a spectral analysis of the generator of the subalgebra. The spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of Jacobi matrices. 1.
Addition formulas for qspecial functions
, 1995
"... Abstract. A general addition formula for a twoparameter family of AskeyWilson polynomials is derived from the quantum SU(2) group theoretic interpretation. This formula contains most of the previously known addition formulas for qLegendre polynomials as special or limiting cases. A survey of the ..."
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Cited by 3 (1 self)
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Abstract. A general addition formula for a twoparameter family of AskeyWilson polynomials is derived from the quantum SU(2) group theoretic interpretation. This formula contains most of the previously known addition formulas for qLegendre polynomials as special or limiting cases. A survey of the literature on addition formulas for qspecial functions using quantum groups and quantum algebras is given. 1. Survey and introduction Many of the wellknown special functions, such as the Jacobi polynomials and Bessel functions, satisfy addition formulas, which can be found in e.g. [1], [12], [63]. Often there exists a group theoretic interpretation of such an addition formula. This means that there exists a group G and a representation t of G in a Hilbert space V such that for a suitable basis {en} of V the matrix elements tn,m: G → C defined by tn,m(g) = 〈t(g)em, en 〉 are known in terms of special functions. Then the homomorphism property (1.1) tn,m(gh) = ∑ tn,p(g)tp,m(h), p
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.