I played around trying to make an equation describing Fibonacci numbers and ended up finding out that what I'd created was something called a recurrence equation:
$f(x)=f(x-1)+f(x-2)$
($f(x)$ is equal to the sum of the function value to the last two previous integer $x$-values's )
I looked it up and found out that this type of recurrence equation can be solved analytically (A solution function exists).
My question goes: Can this type of equation be solved if it contains one or more of the functions derivatives? E.g.: $f(x)=f'(x-1)+f(x-2)$
I can't see why it should be impossible, but I'm unable to find any information on Google and my math teacher doesn't know anything about recurrence equations. Is there a reason why the type of equation (recurrence differential equations) doesn't seem to exist?