One parameter family of linear difference equations and the stability problem for the numerical solution of ODEs

The study of the stability properties of numerical methods leads to considering linear difference equations depending on a complex parameter q. Essentially, the associated characteristic polynomial must have constant type for q ∈ ℂ^-. Usually such request is proved with the help of computers. In this paper, by using the fact that the associated polynomials aresolutions of a "Legendre-type" difference equation, a complete analysis is carried out for the class of linear multistep methods having the highest possible order.