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29
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...
An Algorithmic Proof Theory for Hypergeometric (ordinary and ``$q$'') Multisum/integral Identities
, 1991
"... this paper we show that these fast algorithms can be extended to the much larger class of multisum terminating hypergeometric (or equivalently, binomial coefficient) identities, to constant term identities of DysonMacdonald type, to MehtaDyson type integrals, and more generally, to identities inv ..."
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Cited by 177 (23 self)
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this paper we show that these fast algorithms can be extended to the much larger class of multisum terminating hypergeometric (or equivalently, binomial coefficient) identities, to constant term identities of DysonMacdonald type, to MehtaDyson type integrals, and more generally, to identities involving any (fixed) number of sums and integrals of products of special functions of hypergeometric type. The computergenerated proofs obtained by our algorithms are always short, are often very elegant, and like the singlesum case, sometimes yield the discovery and proof of new identities. We also do the same for single and multi (terminating) qhypergeometric identities, with continuous and/or discrete variables. Here we describe these algorithms in general, and prove their validity. The validity is an immediate consequence of what we call "The fundamental theorem of hypergeometric summation and integration", a result which we believe is of independent theoretical interest and beauty. The technical aspects of our algorithms, as well as their implementation in Maple, will be described in a forthcoming paper. It is possible, and sometimes preferable, to enjoy a magic show without understanding how the tricks are performed. Hence we invite casual readers to go directly to section 6, in which we give several examples of one or two line proofs generated by our method. In order to understand these proofs, and convince oneself of their correctness, one doesn't need to know how they were generated. Readers can generate many more examples on their own once they obtain a copy of our Maple program, that is available upon request from
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].
Variants of the RogersRamanujan Identities
 Adv. in Appl. Math
, 1999
"... We evaluate several integrals involving generating functions of continuous qHermite polynomials in two different ways. The resulting identities give new proofs and generalizations of the RogersRamanujan identities. Two quintic transformations are given, one of which immediately proves the RogersRa ..."
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Cited by 21 (2 self)
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We evaluate several integrals involving generating functions of continuous qHermite polynomials in two different ways. The resulting identities give new proofs and generalizations of the RogersRamanujan identities. Two quintic transformations are given, one of which immediately proves the RogersRamanujan identities without the Jacobi triple product identity. Similar techniques lead to new transformations for unilateral and bilateral series. The quintic transformations lead to curious identities involving primitive 5th roots of unity which are then extended to primitive pth roots of unity for odd p. Running Title: RogersRamanujan Identities Mathematics Subject Classification. Primary 11P82, 33D45, Secondary 42C15. Key words and phrases. qHermite polynomials, RogersRamanujan identities. 1. Introduction. The RogersRamanujan identities 1 X n=0 q n 2 (q; q) n = 1 (q; q 4 ; q 5 ) 1 ; (1.1) 1 X n=0 q n 2 +n (q; q) n = 1 (q 2 ; q 3 ; q 5 ) 1 ; (1.2) play a c...
Generalized rook polynomials and orthogonal polynomials
 IMA Volumes in Mathematics and its Applications
, 1989
"... Abstract. We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials. 1. Introducti ..."
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Cited by 20 (1 self)
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Abstract. We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials. 1. Introduction. Suppose that p0(x), p1(x),... is a sequence of polynomials orthogonal with respect to a measure dµ. Many authors have considered the problem of finding a combinatorial interpretation of the integral � r �
Stationary random fields with linear regressions
 Annals of Probability
, 2001
"... This paper is dedicated to the memory of my friend and collaborator W̷lodzimierz Henryk Smoleński (19521998). We analyze and identify stationary fields with linear regressions and quadratic conditional variances. We give sufficient conditions to determine one dimensional distributions uniquely as n ..."
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Cited by 14 (3 self)
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This paper is dedicated to the memory of my friend and collaborator W̷lodzimierz Henryk Smoleński (19521998). We analyze and identify stationary fields with linear regressions and quadratic conditional variances. We give sufficient conditions to determine one dimensional distributions uniquely as normal, and as certain compactlysupported distributions. Our technique relies on orthogonal polynomials, which under our assumptions turn out to be a version of the so called continuous qHermite polynomials. 1
Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials
 Proc. London Math. Soc
, 1992
"... The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and MeixnerPollaczek polynomials. In particular, we show that Viennot's combina ..."
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Cited by 11 (2 self)
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The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and MeixnerPollaczek polynomials. In particular, we show that Viennot's combinatorial interpretations of the moments can be derived directly from their classical analytical expressions and that the linearization coefficients of MeixnerPollaczek polynomials have an interpretation in the model of derangements analogous to those of Laguerre and Meixner polynomials. 1.
Analytic Combinatorics of Chord Diagrams
 IN FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS (2000
, 2000
"... In this paper we study the enumeration of diagrams of n chords joining 2n points on a circle in disjoint pairs. We establish limit laws for the following three parameters: number of components, size of the largest component, and number of crossings. We also find exact formulas for the moments of the ..."
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Cited by 10 (2 self)
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In this paper we study the enumeration of diagrams of n chords joining 2n points on a circle in disjoint pairs. We establish limit laws for the following three parameters: number of components, size of the largest component, and number of crossings. We also find exact formulas for the moments of the distribution of number of components and number of crossings.
The AskeyWilson polynomials and qSturmLiouville problems
 Math. Proc. Cambridge Philos. Soc. 119
, 1996
"... We nd the adjoint of the AskeyWilson divided di erence operator with respect to the inner product on L 2 (01; 1; (1 0 x 2) 01=2 dx) de ned as a Cauchy principal value and show that the AskeyWilson polynomials are solutions of a qSturmLiouville problem. From these facts we deduce various properti ..."
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Cited by 9 (3 self)
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We nd the adjoint of the AskeyWilson divided di erence operator with respect to the inner product on L 2 (01; 1; (1 0 x 2) 01=2 dx) de ned as a Cauchy principal value and show that the AskeyWilson polynomials are solutions of a qSturmLiouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the AskeyWilson operator.
The combinatorics of qCharlier polynomials
, 1993
"... We describe various aspects of the AlSalamCarlitz qCharlier polynomials. These include combinatorial descriptions of the moments, the orthogonality relation, and the linearization coefficients. ..."
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Cited by 7 (3 self)
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We describe various aspects of the AlSalamCarlitz qCharlier polynomials. These include combinatorial descriptions of the moments, the orthogonality relation, and the linearization coefficients.