$$f(x)=pf(x+1)+qf(x-1)$$
I know how to solve this equation using the generating transform if given a boundary such that $f(a)=0$ and $f(b)=1$. But what if $f(a)=f(b)=0$ ?
I tried using the same method but I have this $f(1)$ in the end that I cannot get rid of, and $f(b)=0$ unused. Should I try a different approach? If so, how should I start?