Common term for differential equations and recurrence relations Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you can generalize these types of equations to equations which involve different "states" or "versions" of a function. Differential equations involve a function and its derivative, and recurrence relations involve a function, and itself in another "state".
Is there a general term for these types of equations, and is there some general results and theory?

They're both examples of the study of finitely-generated modules over a principal ideal domain. In the case of (linear, homogeneous, constant-coefficient) differential equations the PID is $k[D]$ where $D$ is the derivative, and in the case of (linear, homogeneous, constant-coefficient) recurrence relations the PID is $k[x]$ where $x$ is alternately the forward difference, the backward difference, the left shift, or the right shift depending on your preferences. The particular case of the ring $k[x]$ is essentially the theory of Jordan normal form.
