On page 22 of the textbook "Special Functions" of Andrews, el al.
They write that: the factorization:
$$(a)_{2n} = 2^{2n} (\frac{a}{2})_n (\frac{a+1}{2})_n $$
where: $(a)_n := a(a+1)\cdots (a+n-1)$
together with the definition of the gamma function,$\Gamma(x) = \lim_{k\rightarrow \infty} \frac{k! k^{x-1}}{(x)_k}$ leads immediately to Legendre's duplication formula contained in theorem 1.5.1:
$$\Gamma(2a)\Gamma(\frac{1}{2}) = 2^{2a-1} \Gamma(a) \Gamma(a+1/2)$$.
I am attaching scans of what I have done so far, my question is how do I prove this theorem by the appraoch in the book?