On the inverse problem of the discrete calculus of variations

In this paper we present an algorithm to find the discrete Lagrangian for an autonomous recurrence relation of arbitrary even order 2k with k > 1. The method is based on the existence of a set of differential operators called annihilation operators which can be used to convert a functional equation into a system of linear partial differential equations. This completely solves the inverse problem of the calculus of variations in this setting.