(1)

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

(2)

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

(3)

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

(4)

\begin{align*} \frac{1}{C(x+n,n)} & =\sum_{k=0}^{n-1}(-1)^{n-k-1}C(n,k)\frac{n-k}{x+n-k}\\ & =\sum_{k=1}^{n}(-1)^{k-1}C(n,k)\frac{k}{x+k}\cmt{k\rightarrow n-k} \end{align*}