(1)

\begin{align*} \frac{\partial}{\partial z}\zeta\left(s,z\right) & =\frac{\partial}{\partial z}\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{s}}\\ & =\sum_{k=0}^{\infty}\frac{-s}{\left(z+k\right)^{s+1}}\\ & =-s\zeta\left(s+1,z\right) \end{align*}

(2)

\begin{align*} \frac{\partial^{n}}{\partial z^{n}}\zeta\left(s,z\right) & =\left(-s\right)\frac{\partial^{n-1}}{\partial z^{n-1}}\zeta\left(s+1,z\right)\\ & =P\left(-s,n\right)\zeta\left(s+n,z\right) \end{align*}

(3)

\begin{align*} \zeta\left(s,\alpha+\beta\right) & =\sum_{k=0}^{\infty}\frac{\beta^{k}}{k!}\frac{\partial^{k}}{\partial\alpha^{k}}\zeta\left(s,\alpha\right)\\ & =\sum_{k=0}^{\infty}\frac{\beta^{k}}{k!}P\left(-s,k\right)\zeta\left(s+k,\alpha\right)\\ & =\sum_{k=0}^{\infty}\frac{\left(-\beta\right)^{k}}{k!}Q\left(s,k\right)\zeta\left(s+k,\alpha\right)\\ & =\sum_{k=0}^{\infty}\frac{\left(-\beta\right)^{k}}{k!}P\left(s+k-1,k\right)\zeta\left(s+k,\alpha\right)\\ & =\sum_{k=0}^{\infty}\left(-\beta\right)^{k}C\left(s+k-1,k\right)\zeta\left(s+k,\alpha\right) \end{align*}