# Is it possible to convert a higher order difference equation to a system of lower order difference equations?

I know this is possible for differential equations. But is there a similar result for difference equations?

So I was thinking of going from a higher order difference equation describing a single sequence to a system of first order equation describing multiple sequences.

It seems yes. Consider the second order system $$u_{n+1} = u_n+u_{n-1}$$ (i.e. Fibonacci).
Define $$k_n = u_{n-1}$$.
This gives two sequences $$u_n$$ and $$k_n$$. And the system of first order equations governing these is $$u_{n+1} = u_n + k_n$$ and $$k_{n+1} = u_n$$.