\[\sum_{n=0}^{\infty}\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}\left(x%
\right)z^{n}=2\lambda z\left(1-2xz+z^{2}\right)^{-\lambda-1}=\sum_{n=0}^{%
\infty}2\lambda C^{(\lambda+1)}_{n}\left(x\right)z^{n+1},\]