(1)

\begin{align*} \sum_{k=0}^{\infty}P(k,n)x^{k} & =\sum_{k=0}^{\infty}x^{n}\frac{d^{n}}{dx^{n}}x^{k}\\ & =x^{n}\frac{d^{n}}{dx^{n}}\sum_{k=0}^{\infty}x^{k}\\ & =x^{n}\frac{d^{n}}{dx^{n}}\frac{1}{1-x}\\ & =\frac{x^{n}n!}{(1-x)^{n+1}} \end{align*}

(2)

\begin{align*} \sum_{k=0}^{\infty}Q(k,n)x^{k} & =\sum_{k=0}^{\infty}P(k+n-1,n)x^{k}\\ & =\sum_{k=0}^{\infty}x\frac{d^{n}}{dx^{n}}x^{k+n-1}\\ & =x\frac{d^{n}}{dx^{n}}x^{n-1}\sum_{k=0}^{\infty}x^{k}\\ & =x\frac{d^{n}}{dx^{n}}\frac{x^{n-1}}{1-x}\\ & =x\sum_{k=0}^{n}C(n,k)\frac{d^{k}}{dx^{k}}x^{n-1}\frac{d^{n-k}}{dx^{n-k}}\left(1-x\right)^{-1}\\ & =x\sum_{k=0}^{n}C(n,k)P(n-1,k)x^{n-k-1}\left(n-k\right)!\left(1-x\right)^{-(n-k)-1}\\ & =\frac{x^{n}}{\left(1-x\right)^{n+1}}\sum_{k=0}^{n}C(n,k)P(n-1,k)\left(n-k\right)!\left(\frac{1}{x}-1\right)^{k}\\ & =\frac{x^{n}}{\left(1-x\right)^{n+1}}\sum_{k=0}^{n}\frac{n!}{k!(n-k)!}\frac{(n-1)!}{(n-k-1)!}\left(n-k\right)!\left(\frac{1}{x}-1\right)^{k}\\ & =\frac{n!x^{n}}{\left(1-x\right)^{n+1}}\sum_{k=0}^{n}C(n-1,k)\left(\frac{1}{x}-1\right)^{k}\\ & =\frac{n!x^{n}}{\left(1-x\right)^{n+1}}\left(\frac{1}{x}\right)^{n-1}\\ & =\frac{n!}{\left(1-x\right)^{n+1}}x \end{align*}

(3)

\begin{align*} \sum_{k=0}^{\infty}P(\alpha,k)x^{k} & =\sum_{k=0}^{\infty}\frac{\alpha!}{\left(\alpha-k\right)!}x^{k}\\ & =\alpha!x^{\alpha}\sum_{k=0}^{\infty}\frac{1}{\left(\alpha-k\right)!}\left(\frac{1}{x}\right)^{\alpha-k}\\ & =\alpha!\sum_{k=0}^{\infty}\frac{x^{k}}{\left(\alpha-k\right)!}\\ & =\alpha!\sum_{k=0}^{\infty}\frac{\sin\left(\pi(\alpha-k)\right)(1-\alpha+k)!x^{k}}{\pi}\\ & =0 \end{align*}

(4)

\begin{align*} \sum_{k=0}^{\infty}Q(\alpha,k)x^{k} & =\sum_{k=0}^{\infty}\frac{(\alpha+k-1)!}{(\alpha-1)!}x^{k}\\ & =\frac{1}{x^{\alpha-1}(\alpha-1)!}\sum_{k=0}^{\infty}(\alpha+k-1)!x^{\alpha+k-1}\\ & =0 \end{align*}

(5)

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

(6)

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

(7)

\begin{align*} \sum_{k=0}^{\infty}P(k,\alpha)\frac{x^{k}}{k!} & =\sum_{k=0}^{\infty}\frac{k!}{\left(k-\alpha\right)!}\frac{x^{k}}{k!}\\ & =x^{\alpha}\sum_{k=0}^{\infty}\frac{x^{k-\alpha}}{\left(k-\alpha\right)!}\\ & =x^{\alpha}e^{x}\left(1-\frac{\Gamma\left(-\alpha,x\right)}{\Gamma\left(-\alpha\right)}\right) \end{align*}

(8)

\begin{align*} \sum_{k=0}^{\infty}Q(k,\alpha)\frac{x^{k}}{k!} & =\sum_{k=0}^{\infty}\frac{(k+\alpha-1)!}{(k-1)!}\frac{x^{k}}{k!}\\ & =\sum_{k=1}^{\infty}\frac{(k+\alpha-1)!}{(k-1)!}\frac{x^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{(k+\alpha)!}{k!}\frac{x^{k+1}}{(k+1)!}\\ & =x\alpha!\sum_{k=0}^{\infty}\frac{Q(\alpha+1,k)}{Q(2,k)}\frac{x^{k}}{k!}\\ & =x\alpha!F(\alpha+1;2;x) \end{align*}