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

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

Organic Exporter Guide

This guide was developed for the programme 'Export Promotion of Organic Products from Africa' (EPOPA), implemented by Agro Eco and Grolink. The focus of this guide is on export marketing of organic agricultural products. Finished consumer products and other concepts of certification such as Fair Trade and EurepGAP are briefly discussed as well. The guide is written for African exporters starting with organic exports. It may also be useful for business supporters involved in export marketing