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Thread: simplify factorials/gamma's

  1. #1
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    simplify factorials/gamma's

    Hi guys

    I recently ran into a problem after doing an integral, I'm left with $\displaystyle \frac{\beta^{\alpha}}{\Gamma(\alpha)}\frac{\Gamma( \alpha - k)}{\beta^{\alpha-k}}$

    I know that I would be left with $\displaystyle $\beta^{k}$ on the numerator. But in the denominator. I'm not sure how the factorials would cancel.

    Any help is appreciated
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  2. #2
    MHF Contributor
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    Re: simplify factorials/gamma's

    $\dfrac{n!}{(n-k)!} = \underbrace{n(n-1)\cdots (n-k+1)}_{k\text{ terms}}$

    Can you use the Pochhammer rising factorial?
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  3. #3
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    Re: simplify factorials/gamma's

    Are you given that $\displaystyle \alpha$ and k are integers? That is the only way a gamma function can be written as a factorial.
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  4. #4
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    Re: simplify factorials/gamma's

    Hi guys

    thanks for your replies

    yep I'm told for positive integer values $\displaystyle k$ such that $\displaystyle k<\alpha$
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