(1)

\begin{align*} \sum_{k=0}^{\infty}C(x,k)t^{k} & =\sum_{k=0}^{\infty}C(x,k)t^{k}1^{x-k}\\ & =(1+t)^{x} \end{align*}

(2)

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

(3)

\begin{align*} \sum_{x=y}^{\infty}C(x,y)t^{x} & =t^{y}\sum_{s=0}^{\infty}C\left(s+y,y\right)t^{s}\cmt{x=s+y}\\ & =t^{y}\sum_{s=0}^{\infty}P\left(y+s,s\right)\frac{t^{s}}{s!}\\ & =t^{y}\sum_{s=0}^{\infty}Q\left(y+1,s\right)\frac{t^{s}}{s!}\\ & =t^{y}\sum_{s=0}^{\infty}\left(-1\right)^{s}P\left(-\left(y+1\right),s\right)\frac{t^{s}}{s!}\\ & =t^{y}\sum_{s=0}^{\infty}C\left(-\left(y+1\right),s\right)\left(-t\right)^{s}\\ & =t^{y}(1-t)^{-(y+1)} \end{align*}

(4)

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

(5)

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

(6)

\begin{align*} \sum_{x=y}^{\infty}C(x,y)\frac{t^{x}}{x!} & =\frac{1}{y!}\sum_{x=y}^{\infty}\frac{t^{x}}{\left(x-y\right)!}\\ & =\frac{t^{y}}{y!}\sum_{x=0}^{\infty}\frac{t^{x}}{x!}\\ & =\frac{t^{y}}{y!}e^{t} \end{align*}

(7)

\begin{align*} \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}C(j,k)x^{j}y^{k} & =\sum_{j=0}^{\infty}(1+y)^{j}x^{j}\\ & =\sum_{j=0}^{\infty}(x+xy)^{j}\\ & =(1-x-xy)^{-1} \end{align*}

(8)

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

(9)

\begin{align*} \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}C(j+k,k)\frac{x^{j}y^{k}}{(j+k)!} & =\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{x^{j}y^{k}}{j!k!}\\ & =e^{x}e^{y}\\ & =e^{x+y} \end{align*}