What is the difference between Difference equations and Recurrence relations?

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.

• 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." – user26486 Nov 18 '14 at 17:24
• Whether there's a difference will depend on how you define Difference equations. As the Wikipedia article says, definitions vary. – 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.