\[\frac{1+z}{(1-2xz+z^{2})^{\beta+\frac{3}{2}}}=\sum_{n=0}^{\infty}\frac{{\left(%
2\beta+2\right)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\beta+1,\beta)}_{n}\left(%
x\right)z^{n},\]