3,100 research outputs found
-Classical orthogonal polynomials: A general difference calculus approach
It is well known that the classical families of orthogonal polynomials are
characterized as eigenfunctions of a second order linear
differential/difference operator. In this paper we present a study of classical
orthogonal polynomials in a more general context by using the differential (or
difference) calculus and Operator Theory. In such a way we obtain a unified
representation of them. Furthermore, some well known results related to the
Rodrigues operator are deduced. A more general characterization Theorem that
the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn
Tableaux, respectively, is established. Finally, the families of Askey-Wilson
polynomials, q-Racah polynomials, Al-Salam & Carlitz I and II, and q-Meixner
are considered.
[1] R. Alvarez-Nodarse. On characterization of classical polynomials. J.
Comput. Appl. Math., 196:320{337, 2006. [2] M. Alfaro and R. Alvarez-Nodarse. A
characterization of the classical orthogonal discrete and q-polynomials. J.
Comput. Appl. Math., 2006. In press.Comment: 18 page
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
Matrix-Valued Little q-Jacobi Polynomials
Matrix-valued analogues of the little q-Jacobi polynomials are introduced and
studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit
expressions for the orthogonality relations, Rodrigues formula, three-term
recurrence relation and their relation to matrix-valued q-hypergeometric series
and the scalar-valued little q-Jacobi polynomials are presented. The study is
based on a matrix-valued q-difference operator, which is a q-analogue of
Tirao's matrix-valued hypergeometric differential operator.Comment: 16 pages, various corrections and minor additions, incorporating
referee's comment
Multivariable Bessel polynomials related to the hyperbolic Sutherland model with external Morse potential
A multivariable generalisation of the Bessel polynomials is introduced and
studied. In particular, we deduce their series expansion in Jack polynomials, a
limit transition from multivariable Jacobi polynomials, a sequence of
algebraically independent eigenoperators, Pieri type recurrence relations, and
certain orthogonality properties. We also show that these multivariable Bessel
polynomials provide a (finite) set of eigenfunctions of the hyperbolic
Sutherland model with external Morse potential.Comment: a few minor misprints correcte
Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
The annihilation-creation operators are defined as the
positive/negative frequency parts of the exact Heisenberg operator solution for
the `sinusoidal coordinate'. Thus are hermitian conjugate to each
other and the relative weights of various terms in them are solely determined
by the energy spectrum. This unified method applies to most of the solvable
quantum mechanics of single degree of freedom including those belonging to the
`discrete' quantum mechanics.Comment: 43 pages, no figures, LaTeX2e, with amsmath, amssym
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