17,454 research outputs found
Ordered Products, -Algebra, and Two-Variable, Definite-Parity, Orthogonal Polynomials
It has been shown that the Cartan subalgebra of - algebra is the
space of the two-variable, definite-parity polynomials. Explicit expressions of
these polynomials, and their basic properties are presented. Also has been
shown that they carry the infinite dimensional irreducible representation of
the algebra having the spectrum bounded from below. A realization of
this algebra in terms of difference operators is also obtained. For particular
values of the ordering parameter they are identified with the classical
orthogonal polynomials of a discrete variable, such as the Meixner,
Meixner-Pollaczek, and Askey-Wilson polynomials. With respect to variable
they satisfy a second order eigenvalue equation of hypergeometric type. Exact
scattering states with zero energy for a family of potentials are expressed in
terms of these polynomials. It has been put forward that it is the
\.{I}n\"{o}n\"{u}-Wigner contraction and its inverse that form bridge between
the difference and differential calculus.Comment: 17 pages,no figure. to appear in J. Math.Phy
Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement
Two families (type and type ) of confluent hypergeometric polynomials
in several variables are studied. We describe the orthogonality properties,
differential equations, and Pieri type recurrence formulas for these families.
In the one-variable case, the polynomials in question reduce to the Hermite
polynomials (type ) and the Laguerre polynomials (type ), respectively.
The multivariable confluent hypergeometric families considered here may be used
to diagonalize the rational quantum Calogero models with harmonic confinement
(for the classical root systems) and are closely connected to the (symmetric)
generalized spherical harmonics investigated by Dunkl.Comment: AMS-LaTeX v1.2 (with amssymb.sty), 34 page
The semiclassical--Sobolev orthogonal polynomials: a general approach
We say that the polynomial sequence is a semiclassical
Sobolev polynomial sequence when it is orthogonal with respect to the inner
product where is a semiclassical linear functional,
is the differential, the difference or the --difference
operator, and is a positive constant. In this paper we get algebraic
and differential/difference properties for such polynomials as well as
algebraic relations between them and the polynomial sequence orthogonal with
respect to the semiclassical functional . The main goal of this article
is to give a general approach to the study of the polynomials orthogonal with
respect to the above nonstandard inner product regardless of the type of
operator considered. Finally, we illustrate our results by
applying them to some known families of Sobolev orthogonal polynomials as well
as to some new ones introduced in this paper for the first time.Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez
lagomasino on the occasion of his 60th birthday, accepted in Journal of
Approximation Theor
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