I want to solve the following recurrence, for any parameter $x>0$:
$ a[i]= \frac{1}{x} a[i+1]+ (1- \frac{1}{x}) a[i-1]$ for $i \in \{1,...,n-1\}$
$a[0]= 0, a[n]=1$.
i.e. find a closed form for $a[i]$ for any $i \in \{0,...,n\}$
I know that the solution must be $\frac{(x-1)^i-1}{(x-1)^n-1}$,
but I want to find this solution myself. My approach was using the generating function as here, however, I am not sure how to deal with the boundary conditions giving ``on both sides'' i.e. a[0] and a[n].
How can I find the closed solution (using the generating function technique)?