I am trying to solve the following recurrent relation $ T(n)=\frac{n}{m}T(n-1)+\frac{m-n}{m}T(n+1)+1, \,\, \text{subject to } T(m)=0 $
Where $0\leq n\leq m$ and $m$ is a fixed integer. I have written the relation as a difference equation as it is often a way to solve these, and so for $D(n)=T(n)-T(n+1)$ I get $ D(n)=\frac{n}{m-n}D(n-1)+\frac{m}{m-n}. $ Now I want to sum on both side to telescop the sum and use the fact that $T(m)=0$ but my problem is that on the right side, $D(-1)$ is not defined.
The goal of this problem is to find the hitting time from (0,...0) to (1,...,1) for a simple random walk on the $m$-dimensional hypercube and so $T(n)$ represents this (expected) hitting time.
Thanks