Is there any difference between Difference equations and Recurrence relations?
Some people are use them as difference equations and some are use as recurrence relations.
I couldn't find in anywhere. Also I am looking for good reference for first order first degree nonlineardifference equations/Recurrence relations. Thank you.

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    $\begingroup$ Wikipedia: "The term difference equation sometimes refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation." $\endgroup$ – user26486 Nov 18 '14 at 17:24
  • $\begingroup$ Whether there's a difference will depend on how you define Difference equations. As the Wikipedia article says, definitions vary. $\endgroup$ – user26486 Nov 18 '14 at 17:34

There is indeed a difference between difference equations and recurrence relations. In fact, those methods are both equivalent and many books choose to call a relation either a difference one or a recurrence one.

As you may know, a recurrence relation is a relation between terms of a sequence. For example : $ 4*x_{n} = 3*x_{n-1} + 2$. Given a certain number belonging to the sequence, you may find the following one.

A difference equation is defined as :

$ \Delta(a_{n}) = a_{n+1} - a_{n} $ for k = 1

$ \Delta^{2}(a_{n}) = \Delta(a_{n+1}) - \Delta(a_{n}) $ for k = 2

And more generally, $ \Delta^{k}(a_{n}) = \Delta^{k-1}(a_{n+1}) - \Delta^{k-1}(a_{n}) $

So you can also define a series with your difference equations. Most of the time, you can define a series both ways. Just choose the 'syntax' you want to define it. If you choose one, let's say recurrence, you'll be able to switch to a difference equation and vice versa.


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