Formula omitted]-Sparse solutions of linear ordinary differential equations with polynomial coefficients

Abstract

We introduce the notion of m-sparse power series (e.g. expanding sin x and cos x at x=0 gives 2-sparse power series: a coefficient a n of the series can be nonzero only if remainder (n,2) is equal to a fixed number). Then we consider the problem of finding all m-points of a linear ordinary differential equation Ly=0 with polynomial coefficients (i.e., the points at which the equation has a solution in the form of an m-sparse series). It is easy to find an upper bound for m. We prove that if m is fixed then either there exists a finite number of m-points and all of them can be found or all points are m-points and L can be factored as L= L ̃ ∘C where C is an operator of a special kind with constant coefficients. Additionally, we formulate simple necessary and sufficient conditions for the existence of m-points for an irreducible L.