  We develop an approach analogous to classical Frobenius theory for the
analysis of singularities of ODEs in the case of discrete dynamical systems.
Our methodology is based on the Roman-Rota theory of finite operators and
relies crucially on the idea of preserving the Leibniz rule on a lattice of
points by means of the notion of Rota algebras. The relevant cases of the
Bessel, Hermite and Airy equations are discussed.
