Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:
$$ni1=(d1)\sum_{l=1}^{ni1}\frac{\binom{ni1}{l}}{\binom{ni+d3}{l}}$$
Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly: $$ni1=(d1)\sum_{l=1}^{ni1}\frac{\binom{ni1}{l}}{\binom{ni+d3}{l}}$$ 


The canonical reference for this sort of thing is Petkovsek and Zeilberger's book "A=B". Maple (almost certainly) uses the ZeilbergerWilf algorithm for hypergeometric summation (which really goes back to Bill Gosper). You can also read the WilfZeilberger paper (Inventiones, around 1990). 

