\[2n\int_{0}^{x}(1-y)^{\alpha}(1+y)^{\beta}P^{(\alpha,\beta)}_{n}\left(y\right)%
\,\mathrm{d}y=P^{(\alpha+1,\beta+1)}_{n-1}\left(0\right)-(1-x)^{\alpha+1}(1+x)%
^{\beta+1}P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right).\]