Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function.
How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ?
In particular I wonder most about the case $a=1$ and $b=e$. (I already know the trivial cases $a=-1$ and $a=0$)
I know how to express $f(x+1)$ into a taylor series once I have the taylor series for $f(x)$ and I assume this is related ? But does it help to find a closed form solution here ? If there is a closed form solution ? Am I on right track here or do we need to use something completely different or more general ? Does this relate to the fibonacci sequence ?