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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?

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3 Answers 3

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One general approach to unifying differential and difference equations goes by the term time scale calculus. From Wikipedia: "Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set."

Besides the Wikipedia article, a standard reference is Dynamic Equations on Time Scales, by Martin Bohner and Allan Peterson. The field is relatively new - Wikipedia says it was introduced in 1988 by Stefan Hilger in (I believe, although Wikipedia does not say this) his doctoral dissertation - and so it is not that well-known yet.

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Note: "the term" is very misleading. There are various approaches towards such unification, depending on background and emphasis (algebra, analysis, etc). –  Bill Dubuque Jun 2 '11 at 22:45
@Bill: You're right. I'll change it. –  Mike Spivey Jun 2 '11 at 22:47

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.

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Wow, I never thought of it that way (despite the obvious 'operator' form). But can this abstraction (looking at the ring) 1) help you with the solution (i.e. does some knowledge of ring theory help give you solutions), and 2) help you with nonlinear and/or non-homogeneous recurrences/differential equations? –  Mitch Jun 7 '11 at 0:53
1) Yes, once you know what the generalized eigenvectors look like, the theory of Jordan normal form guarantees that solutions are linear combinations of generalized eigenvectors. 2) Non-linear is a completely different ball game but non-homogeneous is doable. In both cases it reduces to finding the homogeneous solutions and a principal solution, and the latter can be done by finding an operator which annihilates the non-homogeneous part. –  Qiaochu Yuan Jun 7 '11 at 10:01
This answer blew my mind. Could you suggest a reference on this? –  Potato Dec 4 '12 at 22:32
@Potato: unfortunately, I did not learn this material from a reference. Once you're familiar with Jordan normal form it is a nice exercise to work out what I claimed above explicitly (in particular once you know what generalized eigenvectors look like you know everything). –  Qiaochu Yuan Dec 4 '12 at 23:04

One approach to unify the theory of differential and difference operators is pseudo-linear algebra - based on work of Ore and Jacobson. For a recent introduction from an algorithmic viewpoint see Manuel Bronstein and Marko Petkovsek: An Introduction to Pseudo-Linear Algebra (1996).

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