1,185 research outputs found

    Properties of some families of hypergeometric orthogonal polynomials in several variables

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    Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the norms of the polynomials in terms of the norm of the constant polynomial.Comment: 42 pages, AMSLaTeX 1.1 with amssym

    Quadratic transformations for orthogonal polynomials in one and two variables

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    We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or qq-Askey scheme. In the two-variable case we focus, after some generalities, on the polynomials associated with root system BC2BC_2, i.e., BC2BC_2-type Jacobi polynomials if q=1q=1 and Koornwinder polynomials in two variables in the qq-case.Comment: v2: minor corrections, 24 page

    Convolutions for orthogonal polynomials from Lie and quantum algebra representations

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    The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to generalisations of the convolution identities for these polynomials. Using the Racah coefficients convolution identities for continuous Hahn, Hahn and Jacobi polynomials are obtained. From the quantised universal enveloping algebra for su(1,1) convolution identities for the Al-Salam and Chihara polynomials and the Askey-Wilson polynomials are derived by using the Clebsch-Gordan and Racah coefficients. For the quantised universal enveloping algebra for su(2) q-Racah polynomials are interpreted as Clebsch-Gordan coefficients, and the linearisation coefficients for a two-parameter family of Askey-Wilson polynomials are derived.Comment: AMS-TeX, 31 page

    Orthogonal polynomial kernels and canonical correlations for Dirichlet measures

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    We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet multinomial distribution, respectively, on the continuous and the N-discrete d-dimensional simplex. Their infinite-dimensional limit distributions, respectively, the Poisson-Dirichlet distribution and Ewens's sampling formula, are considered as well. We study, in particular, the possibility of mapping canonical correlations on the d-dimensional continuous simplex (i) to canonical correlation sequences on the d+1-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and vice versa. Driven by this motivation, the first half of the paper is devoted to providing a full characterization and probabilistic interpretation of n-orthogonal polynomial kernels (i.e., sums of products of orthogonal polynomials of the same degree n) with respect to the mentioned marginal distributions. We establish several identities and some integral representations which are multivariate extensions of important results known for the case d=2 since the 1970s. These results, along with a common interpretation of the mentioned kernels in terms of dependent Polya urns, are shown to be key features leading to several non-trivial solutions to Lancaster's problem, many of which can be extended naturally to the limit as dd\rightarrow\infty.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ403 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

    Limit relations between qq-Krall type orthogonal polynomials

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    In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a qq-classical linear functional via the addition of one or two mass points. The limit relations between the qq-Krall type modification of big qq-Jacobi, little qq-Jacobi, big qq-Laguerre, and other families of the qq-Hahn tableau are established.Comment: 19 Pages, 3 tables, 1 figur

    8 Lectures on quantum groups and q-special functions

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    Lecture notes for an eight hour course on quantum groups and qq-special functions at the fourth Summer School in Differential Equations and Related Areas, Universidad Nacional de Colombia and Universidad de los Andes, Bogot\'a, Colombia, July 22 -- August 2, 1996. The lecture notes contain an introduction to quantum groups, qq-special 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 Askey-Wilson polynomials out in detail as the main example. As an application we derive an addition formula for a two-parameter subfamily of Askey-Wilson polynomials. A relation between the Al-Salam and Chihara polynomials and the quantised universal enveloping algebra for su(1,1)su(1,1) is given. Finally, more examples and other approaches as well as some open problems are given.Comment: AMS-TeX, 82 page

    Some classical multiple orthogonal polynomials

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    Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system of several functions. We describe seven families of multiple orthogonal polynomials which have he same flavor as the very classical orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some open research problems and some applications
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