Prove that .

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- May 3rd 2008, 08:44 AMjames_bondSums
Prove that .

- May 3rd 2008, 09:35 AMIsomorphism
I could not prove this, but I have some lead:

Strangely I get

I first observed that

But is the coefficient of in

Viewed differently: The coefficient of in i !!

To get that elusive , I tried the following(in vain)

But what goes on from here is pretty ugly (Crying) - May 3rd 2008, 01:13 PMPaulRS
Let's find the generating functions

Note that:

Thus:

On the other hand:

Thus:

SO we get that:

That is, the generating functions are equal, therefore we must have:

- November 1st 2008, 05:30 AMPaulRS
First note that, given , we have

From there it follows that, given we have:

Now, let's go to the sums.

The LHS is equal to: . by the linearity of the integral:

And by the Binomial Theorem this equals:

But by the Binomial Theorem again: and now using the linearity of the integral we get and we are done

**Comment:**In general where is a contour enclosing 0

**No animals were harmed during the production of this solution (Rofl)**