321 research outputs found
A Bochner Theorem for Dunkl Polynomials
We establish an analogue of the Bochner theorem for first order operators of
Dunkl type, that is we classify all such operators having polynomial solutions.
Under natural conditions it is seen that the only families of orthogonal
polynomials in this category are limits of little and big -Jacobi
polynomials as
A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials
A novel family of orthogonal polynomials called the Chihara polynomials
is characterized. The polynomials are obtained from a "continuous" limit of the
complementary Bannai-Ito polynomials, which are the kernel partners of the
Bannai-Ito polynomials. The three-term recurrence relation and the explicit
expression in terms of Gauss hypergeometric functions are obtained through a
limit process. A one-parameter family of second-order differential Dunkl
operators having these polynomials as eigenfunctions is also exhibited. The
quadratic algebra with involution encoding this bispectrality is obtained. The
orthogonality measure is derived in two different ways: by using Chihara's
method for kernel polynomials and, by obtaining the symmetry factor for the
one-parameter family of Dunkl operators. It is shown that the polynomials are
related to the big Jacobi polynomials by a Christoffel transformation and
that they can be obtained from the big -Jacobi by a limit.
The generalized Gegenbauer/Hermite polynomials are respectively seen to be
special/limiting cases of the Chihara polynomials. A one-parameter extension of
the generalized Hermite polynomials is proposed
Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
Based on the theory of Dunkl operators, this paper presents a general concept
of multivariable Hermite polynomials and Hermite functions which are associated
with finite reflection groups on \b R^N. The definition and properties of
these generalized Hermite systems extend naturally those of their classical
counterparts; partial derivatives and the usual exponential kernel are here
replaced by Dunkl operators and the generalized exponential kernel K of the
Dunkl transform. In case of the symmetric group , our setting includes the
polynomial eigenfunctions of certain Calogero-Sutherland type operators. The
second part of this paper is devoted to the heat equation associated with
Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem
is governed by a positive one-parameter semigroup; this is assured by a maximum
principle for the generalized Laplacian. The explicit solution to the Cauchy
problem involves again the kernel K, which is, on the way, proven to be
nonnegative for real arguments.Comment: 24 pages, AMS-LaTe
A "missing" family of classical orthogonal polynomials
We study a family of "classical" orthogonal polynomials which satisfy (apart
from a 3-term recurrence relation) an eigenvalue problem with a differential
operator of Dunkl-type. These polynomials can be obtained from the little
-Jacobi polynomials in the limit . We also show that these polynomials
provide a nontrivial realization of the Askey-Wilson algebra for .Comment: 20 page
Bispectrality of the Complementary Bannai-Ito Polynomials
A one-parameter family of operators that have the complementary Bannai-Ito
(CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the
kernel partners of the Bannai-Ito polynomials and also correspond to a
limit of the Askey-Wilson polynomials. The eigenvalue
equations for the CBI polynomials are found to involve second order Dunkl shift
operators with reflections and exhibit quadratic spectra. The algebra
associated to the CBI polynomials is given and seen to be a deformation of the
Askey-Wilson algebra with an involution. The relation between the CBI
polynomials and the recently discovered dual -1 Hahn and para-Krawtchouk
polynomials, as well as their relation with the symmetric Hahn polynomials, is
also discussed
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