(1)

\begin{align*} C\left(\alpha,\beta\right)C\left(\beta,\gamma\right) & =\frac{\alpha!}{\beta!\left(\alpha-\beta\right)!}\cdot\frac{\beta!}{\gamma!\left(\beta-\gamma\right)!}\\ & =\frac{\alpha!}{\left(\alpha-\beta\right)!\gamma!\left(\beta-\gamma\right)}\\ & =\frac{\alpha!}{\gamma!\left(\alpha-\gamma\right)!}\cdot\frac{\left(\alpha-\gamma\right)!}{\left(\alpha-\beta\right)!\left(\beta-\gamma\right)}\\ & =C\left(\alpha,\gamma\right)C\left(\alpha-\gamma,\beta-\gamma\right) \end{align*}

(2)

\begin{align*} \left(-1\right)^{k}C\left(z,k\right) & =\left(-1\right)^{k}\frac{P\left(z,k\right)}{k!}\\ & =\frac{Q\left(-z,k\right)}{k!}\\ & =\frac{P\left(k-z-1,k\right)}{k!}\\ & =C\left(k-z-1,k\right) \end{align*}