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Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly:

$$n-i-1=(d-1)\sum_{l=1}^{n-i-1}\frac{\binom{n-i-1}{l}}{\binom{n-i+d-3}{l}}$$

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I recommend rewriting your question so that it makes more sense. Writing t for n-i-1 would be a good start, as well as asking what you really want, rather than a reference to why Maple does things correctly. Gerhard "Ask Me About System Design" Paseman, 2011.11.30 –  Gerhard Paseman Nov 30 '11 at 16:50
    
I don't get your question: do you mean that you are surprised that Maple can do something correctly? Or, rather than why, are you asking how does Maple derive the identity. $$ $$ Anyway, the question is not clear as is, it needs motivation. See: mathoverflow.net/howtoask –  Thierry Zell Nov 30 '11 at 17:20
    
A possible explanation, a bit tautological: Maple is able to make a number of formal simplifications on expressions like yours, so as to reduce them to some of the thousands of standard identities that it contains in its memory. –  Pietro Majer Nov 30 '11 at 17:25
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You can get Maple to show the steps it is taking . That may or may not give a satisfying result (if that is what you really want to know). Improve your notation and try the cases $d=1,2,3,4$ to get some insight. –  Aaron Meyerowitz Nov 30 '11 at 17:52
    
I am curious about why anyone would down vote this?! It is a perfectly reasonable question. –  Igor Rivin Nov 30 '11 at 19:14
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up vote 3 down vote accepted

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).

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