619 research outputs found
The Jacobi inversion formula
We look for spectral type differential equations satisfied by the generalized
Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to
a weight function consisting of the classical Jacobi weight function together
with point masses at the endpoints of the interval of orthogonality.
In order to find explicit formulas for the coefficients of these differential
equations we have to solve systems of equations involving derivatives of the
classical Jacobi polynomials. These systems of equations have a unique solution
which can be given explicitly in terms of Jacobi polynomials. This is a
consequence of the Jacobi inversion formula which is proved in this paper.Comment: 15 page
On differential equations for Sobolev-type Laguerre polynomials
We obtain all spectral type differential equations satisfied by the
Sobolev-type Laguerre polynomials. This generalizes the results found in 1990
by the first and second author in the case of the generalized Laguerre
polynomials defined by T.H. Koornwinder in 1984.Comment: 45 page
Differential equations for generalized Jacobi polynomials
We look for spectral type differential equations satisfied by the generalized
Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to
a weight function consisting of the classical Jacobi weight function together
with two point masses at the endpoints of the interval of orthogonality.
We show that such a differential equation is uniquely determined and we give
explicit representations for the coefficients.
In case of nonzero mass points the order of this differential equation is
infinite, except for nonnegative integer values of (one of) the parameters.
Otherwise, the finite order is explictly given in terms of the parameters.Comment: 33 pages, submitted for publicatio
Epicycles and Poincar\'{e} Resonances in General Relativity
The method of geodesic deviations provides analytic approximations to
geodesics in arbitrary background space-times. As such the method is a useful
tool in many practical situations. In this note we point out some subtleties in
the application of the method related to secular motions, in first as well as
in higher order. In particular we work out the general second-order
contribution to bound orbits in Schwarzschild space-time and show that it
provides very good analytical results all the way up to the innermost stable
circular orbit.Comment: 24 pages, 4 figure
Transformation design and nonlinear Hamiltonians
We study a class of nonlinear Hamiltonians, with applications in quantum
optics. The interaction terms of these Hamiltonians are generated by taking a
linear combination of powers of a simple `beam splitter' Hamiltonian. The
entanglement properties of the eigenstates are studied. Finally, we show how to
use this class of Hamiltonians to perform special tasks such as conditional
state swapping, which can be used to generate optical cat states and to sort
photons.Comment: Accepted for publication in Journal of Modern Optic
Limits of elliptic hypergeometric biorthogonal functions
The purpose of this article is to bring structure to (basic) hypergeometric
biorthogonal systems, in particular to the q-Askey scheme of basic
hypergeometric orthogonal polynomials. We aim to achieve this by looking at the
limits as p->0 of the elliptic hypergeometric biorthogonal functions from
Spiridonov, with parameters which depend in varying ways on p. As a result we
get 38 systems of biorthogonal functions with for each system at least one
explicit measure for the bilinear form. Amongst these we indeed recover the
q-Askey scheme. Each system consists of (basic hypergeometric) rational
functions or polynomials.Comment: 27 pages. This is a self-contained article which can also be seen as
part 1 of a 3 part series on limits of (multivariate) elliptic hypergeometric
biorthogonal functions and their measure
Interpolation of SUSY quantum mechanics
Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics
, is discussed together
with related operators. For a wide variety of shape-invariant degree one
quantum mechanics and their `discrete' counterparts, the interpolation
Hamiltonian is also shape-invariant, that is it takes the same form as the
original Hamiltonian with shifted coupling constant(s).Comment: 18 page
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