Interesting Combinatorial Sum
Point of post: This is just an interesting combinatorial sum I solved on AOPS a few minutes ago. I mention it because, while sometimes less elegant, one must remember to never disregard “continuous” math even when one is working in “discrete” areas.
Problem: For such that prove that
Proof: The first thing one might notice is that
But, rearranging this we arrive at
but appealing to the Binomial Theorem we see that the above is equal to
where is the Beta Function. But, appealing to an alternative form of the Beta function we see that the sum is equal to
where is the Gamma Function.
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