1,185 research outputs found
Properties of some families of hypergeometric orthogonal polynomials in several variables
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
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 -Askey scheme. In
the two-variable case we focus, after some generalities, on the polynomials
associated with root system , i.e., -type Jacobi polynomials if
and Koornwinder polynomials in two variables in the -case.Comment: v2: minor corrections, 24 page
Convolutions for orthogonal polynomials from Lie and quantum algebra representations
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
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 .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 -Krall type orthogonal polynomials
In this paper, we consider a natural extension of several results related to
Krall-type polynomials introducing a modification of a -classical linear
functional via the addition of one or two mass points. The limit relations
between the -Krall type modification of big -Jacobi, little -Jacobi,
big -Laguerre, and other families of the -Hahn tableau are established.Comment: 19 Pages, 3 tables, 1 figur
8 Lectures on quantum groups and q-special functions
Lecture notes for an eight hour course on quantum groups and -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, -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 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
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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