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Can I simplify the sum given below further so as to avoid computation using large numbers, $$\sum_{k=0}^b {{a+k} \choose k} {{c-a-1+d-k} \choose {d-k}} $$

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Are there any requirements on $c,a,d,b$? –  JSchlather Dec 6 '12 at 15:28
@JacobSchlather a,b,c,d are positive integers with b<d and a<c. –  user52032 Dec 6 '12 at 15:37
This almost looks like the Vandermonde identity, but the sum isn't complete. Looking at the proof of that identity might give you something, though. –  Jonathan Christensen Dec 6 '12 at 16:38
If you indeed have $b< d$ instead of the other way around, then you cannot simplify it. –  Phira Dec 6 '12 at 16:50
@JonathanChristensen thanks for the input, I will look into that –  user52032 Dec 6 '12 at 16:56

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