I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed that transformation of variables is one of the most common approaches to solve a functional equation.
Example: If we have a functional equation (polynomial with real coefficients):
$f(x+1)-f(x)=f(x)-f(x-1)+2$ (I). Then it might be a good idea to let:
so that our equation can be written:
But I have noticed that (I) actually can be written as:
$h(x+1)=h(x)+2$ and I can't see why. If we let $t=x-1$ we can write the right hand side of our equation as $f(t+1)-f(t)+2=h(t)+2$ but I can't see why the left hand side becomes $h(x+1)$ instead of $h(x)$.
My question more specifically:
How may $f(x+1)-f(x)=f(x)-f(x-1)+2$ be written as $h(x+1)=h(x)+2$