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Fun Proof by Induction
Posted:
Jul 18, 1996 2:27 PM


My previous posting to this group has gone unanswered, no doubt, partly due to the complexity of my setting it up. So, here is a special case of the following.
Fix positive integer p >=2. Let G= power set of {1,2,...p1}. Let g be any element of G. Define g to be even if its size is even, odd if its size is odd.
Let F(x,p) = sum over even g in G of product over i in G of (x+pi)choosex
(The binomial coefficient).
Let H(x,p) = sum over odd g in G of product over i in G of (x+pi)choosex.
Let D(x,p) = F(x,p)  H(x,p)
I have already verified for p=4 and p=5 what D(x,p) is. The obvious conjecture derived from these cases is that
D(x,p) = (1)^p/(p!) times Product of (x+2 i) from i = 1 through p.
I have tried to prove this conjecture by induction but have not succeeded. Please help. Feel free to email me. Thank you. John



