\[\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\,\mathrm{d}x=\frac{(-1)^{n}{\left(%
\frac{1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}z\right)_{n+1}}},\]