Rational Binomial Identity Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:
$$n-i-1=(d-1)\sum_{l=1}^{n-i-1}\frac{\binom{n-i-1}{l}}{\binom{n-i+d-3}{l}}$$
 A: The canonical reference for this sort of thing is Petkovsek and Zeilberger's book "A=B". Maple (almost certainly) uses the Zeilberger-Wilf algorithm for hypergeometric summation (which really goes back to Bill Gosper). You can also read the Wilf-Zeilberger paper (Inventiones, around 1990).
A: As Gerhard Paseman suggested, it look better to replace $m=n-i-1$ and also $x=d-2$. With this the question takes the form $S:=\sum_{\ell=1}^m\binom{m}{\ell}\binom{m+x}{\ell}^{-1}\frac{x+1}m$ and we show $S\equiv1$. Let
$$F(m,\ell):=\binom{m}{\ell}\binom{m+x}{\ell}^{-1}\frac{x+1}m \qquad \text{and}
\qquad
G(m,\ell):=-\binom{m}{\ell}\binom{m+x+1}{\ell}^{-1}\frac{m+x+1}m.$$
It's easy to check 
$$F(m,\ell)=G(m,\ell+1)-G(m,\ell), \tag1$$
say by dividing through with $F(m,\ell)$ and simplifying. Summing (1) over integers $\ell\geq0$ gives
$$\sum_{\ell=0}^mF(m,\ell)
=\sum_{\ell\geq0}G(m,\ell+1)-\sum_{\ell\geq0}G(m,\ell)=-G(m,0)=\frac{x+m+1}m.$$
Therefore, we arrive at
$$S=\sum_{\ell=1}^mF(m,\ell)=\sum_{\ell=0}^mF(m,\ell)-F(m,0)=\frac{x+m+1}m-\frac{x+1}m=1$$
which is what aimed for.
