(1)

\begin{align*} P\left(-\frac{1}{2},n\right) & =\frac{\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(1-\left(\frac{1}{2}+n\right)\right)}\\ & =\frac{1}{\pi}\sin\left(\left(\frac{1}{2}+n\right)\pi\right)\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{2}+n\right)\\ & =\frac{(-1)^{n}}{\pi}\Gamma\left(\frac{1}{2}\right)\frac{(2\pi)^{\frac{1}{2}}\Gamma(2n)}{2^{2n-\frac{1}{2}}\Gamma(n)}\\ & =\frac{(-1)^{n}\Gamma(2n)}{2^{2n-1}\Gamma(n)}\\ & =\frac{(-1)^{n}(2n-1)!}{2^{2n-1}(n-1)!} \end{align*}

(1)-2

\begin{align*} P\left(-\frac{1}{2},n\right) & =\prod_{j=0}^{n-1}\left(-\frac{1}{2}-j\right)\\ & =\prod_{j=0}^{n-1}\left(-\frac{1}{2}\left(1+2j\right)\right)\\ & =\left(-\frac{1}{2}\right)^{n}(2n-1)!!\\ & =\left(-\frac{1}{2}\right)^{n}\frac{(2n-1)!}{(2n-2)!!}\\ & =\left(-\frac{1}{2}\right)^{n}\frac{(2n-1)!}{2^{n-1}(n-1)!}\\ & =\frac{(-1)^{n}(2n-1)!}{2^{2n-1}(n-1)!} \end{align*}

(2)

\begin{align*} Q\left(\frac{1}{2},n\right) & =P\left(-\frac{1}{2}+n,n\right)\\ & =(-1)^{n}P\left(-\frac{1}{2},n\right)\\ & =\frac{(2n-1)!}{2^{2n-1}(n-1)!} \end{align*}