Formal Theory for Differential-Difference Operators

Differential-difference operators are linear operators involving both d/dz and the shift z ↦ z + 1 (or z(d/dz) and z ↦ qz). The aim is to give a formal classification and to provide solutions for these equations. Differential-difference operators can be considered as formal differential operators of infinite order. For the latter one studies Newton polygons, factorizations, solutions and developes a theory of symbolic solutions. This theory applied to differential-difference operators seems in many case adequate. In other cases, one cannot produce enough symbolic solutions. Independent from differential operators of infinite order, certain systems of differential-difference are treated. Here the theory seems complete. Many examples illustrate the theory.