I was just wondering how I would evaluate something like:
$\displaystyle {\frac{1}{2} \choose 1}$ (1/2 choose 1)
or
$\displaystyle {\frac{1}{2} \choose 2}$ (1/2 choose 2)
I've never seen fractions in binomial coefficients.
I was just wondering how I would evaluate something like:
$\displaystyle {\frac{1}{2} \choose 1}$ (1/2 choose 1)
or
$\displaystyle {\frac{1}{2} \choose 2}$ (1/2 choose 2)
I've never seen fractions in binomial coefficients.
Wikipedia
Binomial coefficient - Wikipedia, the free encyclopedia
talks about that way down the page at "Binomial Coefficient with n= 1/2".
Essentially, you write $\displaystyle \begin{pmatrix}n \\ m\end{pmatrix}$ as $\displaystyle \frac{n!}{m!(n-m)!}= \frac{n(n-1)(n- 2)\cdot\cdot\cdot(n-m+1)}{m(m-1)(m-2)\cdot\cdot\cdot(3)(2)(1)}$ 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,
$\displaystyle \begin{pmatrix}\frac{1}{2} \\ n \end{pmatrix}= \begin{pmatrix}2k+1 \\ k\end{pmatrix}\frac{(-1)^{k+1}(k+1)}{2^{2n}(2k-1)(2k+1)}$