14,388 research outputs found
Semi-classical Laguerre polynomials and a third order discrete integrable equation
A semi-discrete Lax pair formed from the differential system and recurrence
relation for semi-classical orthogonal polynomials, leads to a discrete
integrable equation for a specific semi-classical orthogonal polynomial weight.
The main example we use is a semi-classical Laguerre weight to derive a third
order difference equation with a corresponding Lax pair.Comment: 11 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
Orthogonal Polynomials from Hermitian Matrices II
This is the second part of the project `unified theory of classical
orthogonal polynomials of a discrete variable derived from the eigenvalue
problems of hermitian matrices.' In a previous paper, orthogonal polynomials
having Jackson integral measures were not included, since such measures cannot
be obtained from single infinite dimensional hermitian matrices. Here we show
that Jackson integral measures for the polynomials of the big -Jacobi family
are the consequence of the recovery of self-adjointness of the unbounded Jacobi
matrices governing the difference equations of these polynomials. The recovery
of self-adjointness is achieved in an extended Hilbert space on which
a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a
difference Schr\"odinger operator for an infinite dimensional eigenvalue
problem. The polynomial appearing in the upper/lower end of Jackson integral
constitutes the eigenvector of each of the two unbounded Jacobi matrix of the
direct sum. We also point out that the orthogonal vectors involving the
-Meixner (-Charlier) polynomials do not form a complete basis of the
Hilbert space, based on the fact that the dual -Meixner polynomials
introduced in a previous paper fail to satisfy the orthogonality relation. The
complete set of eigenvectors involving the -Meixner polynomials is obtained
by constructing the duals of the dual -Meixner polynomials which require the
two component Hamiltonian formulation. An alternative solution method based on
the closure relation, the Heisenberg operator solution, is applied to the
polynomials of the big -Jacobi family and their duals and -Meixner
(-Charlier) polynomials.Comment: 65 pages. Comments, references and table of contents are added. To
appear in J.Math.Phy
Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy
A -semi-classical weight is one which satisfies a particular
linear, first order homogeneous equation in a divided-difference operator
. It is known that the system of polynomials, orthogonal with
respect to this weight, and the associated functions satisfy a linear, first
order homogeneous matrix equation in the divided-difference operator termed the
spectral equation. Attached to the spectral equation is a structure which
constitutes a number of relations such as those arising from compatibility with
the three-term recurrence relation. Here this structure is elucidated in the
general case of quadratic lattices. The simplest examples of the
-semi-classical orthogonal polynomial systems are precisely those
in the Askey table of hypergeometric and basic hypergeometric orthogonal
polynomials. However within the -semi-classical class it is
entirely natural to define a generalisation of the Askey table weights which
involve a deformation with respect to new deformation variables. We completely
construct the analogous structures arising from such deformations and their
relations with the other elements of the theory. As an example we treat the
first non-trivial deformation of the Askey-Wilson orthogonal polynomial system
defined by the -quadratic divided-difference operator, the Askey-Wilson
operator, and derive the coupled first order divided-difference equations
characterising its evolution in the deformation variable. We show that this
system is a member of a sequence of classical solutions to the
-Painlev\'e system.Comment: Submitted to Duke Mathematical Journal on 5th April 201
Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials
Eigenvalues and eigenfunctions of the volume operator, associated with the
symmetric coupling of three SU(2) angular momentum operators, can be analyzed
on the basis of a discrete Schroedinger-like equation which provides a
semiclassical Hamiltonian picture of the evolution of a `quantum of space', as
shown by the authors in a recent paper. Emphasis is given here to the
formalization in terms of a quadratic symmetry algebra and its automorphism
group. This view is related to the Askey scheme, the hierarchical structure
which includes all hypergeometric polynomials of one (discrete or continuous)
variable. Key tool for this comparative analysis is the duality operation
defined on the generators of the quadratic algebra and suitably extended to the
various families of overlap functions (generalized recoupling coefficients).
These families, recognized as lying at the top level of the Askey scheme, are
classified and a few limiting cases are addressed.Comment: 10 pages, talk given at "Physics and Mathematics of Nonlinear
Phenomena" (PMNP2013), to appear in J. Phys. Conf. Serie
The Symmetrical -Semiclassical Orthogonal Polynomials of Class One
We investigate the quadratic decomposition and duality to classify
symmetrical -semiclassical orthogonal -polynomials of class one where
is the Hahn's operator. For any canonical situation, the recurrence
coefficients, the -analog of the distributional equation of Pearson type,
the moments and integral or discrete representations are given
Bannai-Ito polynomials and dressing chains
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG
maps are related to dressing chains determined by quadratic algebras. The
Bannai-Ito polynomials and their kernel polynomials -- the complementary
Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.Comment: 15 pages; Section 2 is slightly modified and a few typos are
correcte
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