Explicit representations of biorthogonal polynomials

Given a parametrised weight function $\omega(x,\mu)$ such that the quotients of its consecutive moments are M\"obius maps, it is possible to express the underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present paper we address ourselves to two related issues. Firstly, we demonstrate that, subject to additional assumptions, every such $\omega$ obeys (in $x$) a linear differential equation whose solution is a generalized hypergeometric function. Secondly, using a generalization of standard divided differences, we present a new explicit representation of the underlying orthogonal polynomials

The Numerical Solution of Differential and Differential-Algebraic Systems

Systems of ordinary differential equations (ODE) or ordinary differential/algebraic equations (DAE) are well-known mathematical models. The numerical solution of such systems are discussed. For (ODE) we mention some available codes and stress the need of type insensitive versions. Further the term stiffness is redefined, and ideas on handling discontinuities are presented. The paper ends with a discussion of index for DAE

Regarding the absolute stability of Störmer-Cowell methods

Störmer-Cowell methods, a popular class of methods for computations in celestial mechanics, is known to exhibit orbital instabilities when the order of the methods exceed two. Analysing the absolute stability of Störmer-Cowell methods close to zero we present a characterization of these instabilities for methods of all orders.nrpages: 14status: publishe

Efficient Quadrature of Highly Oscillatory Integrals Using Derivatives

In this paper we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome are two families of methods, one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of Filon. Both kinds of methods approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency. We determine asymptotic properties of these methods, proving that, perhaps counterintuitively, their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas