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Math Help - Evaluting Binomial Coefficients that contain fractions?

  1. #1
    Member eXist's Avatar
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    Evaluting Binomial Coefficients that contain fractions?

    I was just wondering how I would evaluate something like:

    {\frac{1}{2} \choose 1} (1/2 choose 1)
    or
    {\frac{1}{2} \choose 2} (1/2 choose 2)

    I've never seen fractions in binomial coefficients.
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  2. #2
    MHF Contributor

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    Wikipedia
    Binomial coefficient - Wikipedia, the free encyclopedia
    talks about that way down the page at "Binomial Coefficient with n= 1/2".

    Essentially, you write \begin{pmatrix}n \\ m\end{pmatrix} as \frac{n!}{m!(n-m)!}= \frac{n(n-1)(n- 2)\cdot\cdot\cdot(n-m+1)}{m(m-1)(m-2)\cdot\cdot\cdot(3)(2)(1)} with the difference that since n is not an integer, you never end at "1" or "n- m+1". Instead, you get an infinite product which must be shown to have a finite value.

    Wikipedia gives, explicitely,
    \begin{pmatrix}\frac{1}{2} \\ n \end{pmatrix}= \begin{pmatrix}2k+1 \\ k\end{pmatrix}\frac{(-1)^{k+1}(k+1)}{2^{2n}(2k-1)(2k+1)}
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