LINEAR ODE’S WITH LAURENT POLYNOMIAL COEFFICIENTS

Abstract

The forward and backward shift operators (and hence all Laurent polynomials in these operators) act on two-sided infinite sequences of continuous functions as well as the differentiation operator. One can define therefore linear ODEs with Laurent polynomial coefficients, where the unknowns are such sequences. It turns out that equations of this type can be treated easily, exactly as linear ODEs with constant coefficients. Our motivation for considering these ODEs, which seem to be quite natural on their own, has been the fact that the collection of all Bessel functions is characterized by a very simple first order equation of this kind.