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

Interpolation of SUSY quantum mechanics

Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics $H_s=(1-s)A^{\dagger}A + sAA^{\dagger}$, $0\le s\le 1$ 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

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