Integral Representation of Gamma Function

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April 4th 2010, 04:15 AM
EinStone

Integral Representation of Gamma Function

Let $\Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}$ .

Show, for every $N \in \mathbb{N}$ that

$\int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s)$ .

Hint: For the second inequality, divide both sides by $N^s$ and compare the resulting rational funtions in terms of their poles.
April 4th 2010, 10:24 AM
chiph588@

Quote:

Originally Posted by EinStone

Let $\Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}$ .

Show, for every $N \in \mathbb{N}$ that

$\int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s)$ .

Hint: For the second inequality, divide both sides by $N^s$ and compare the resulting rational funtions in terms of their poles.

Here's a hint for the first equality:

$\left(1-\frac{t}{N}\right)^N = \sum_{k=0}^N {N\choose k} \left(-\frac{t}{N}\right)^k$ by the binomial expansion theorem.
April 5th 2010, 01:27 AM
EinStone

I don't see how this can help me... :( , can you give further help?

EDIT: Ah lol, I just saw that I can pull everything out of the integral then its easy. Ok I'll now try the second equation, if you can give any help please do it :).

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