12,680 research outputs found
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
Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States
In this paper, we study a family of orthogonal polynomials
arising from nonlinear coherent states in quantum optics. Based on the
three-term recurrence relation only, we obtain a uniform asymptotic expansion
of as the polynomial degree tends to infinity. Our asymptotic
results suggest that the weight function associated with the polynomials has an
unusual singularity, which has never appeared for orthogonal polynomials in the
Askey scheme. Our main technique is the Wang and Wong's difference equation
method. In addition, the limiting zero distribution of the polynomials
is provided
Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function
Laurent polynomials related to the Hahn-Exton -Bessel function, which are
-analogues of the Lommel polynomials, have been introduced by Koelink and
Swarttouw. The explicit strong moment functional with respect to which the
Laurent -Lommel polynomials are orthogonal is given. The strong moment
functional gives rise to two positive definite moment functionals. For the
corresponding sets of orthogonal polynomials the orthogonality measure is
determined using the three-term recurrence relation as a starting point. The
relation between Chebyshev polynomials of the second kind and the Laurent
-Lommel polynomials and related functions is used to obtain estimates for
the latter
Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction
We study a family of the Laurent biorthogonal polynomials arising from the
Hermite continued fraction for a ratio of two complete elliptic integrals.
Recurrence coefficients, explicit expression and the weight function for these
polynomials are obtained. We construct also a new explicit example of the
Szeg\"o polynomials orthogonal on the unit circle. Relations with associated
Legendre polynomials are considered.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
- …