(1)

\begin{align*} P(x,y) & =\frac{x!}{(x-y)!}\\ & =\frac{x!}{(x-y)!}\frac{(x+1)-(x-y)}{1+y}\\ & =\frac{1}{y+1}\left(\frac{(x+1)!}{(x-y)!}-\frac{x!}{(x-y-1)!}\right)\\ & =\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right) \end{align*}

(1)-2

\begin{align*} P(x,y) & =\frac{1}{(x+1)(x-y)}P(x+1,y+2)\\ & =\frac{1}{y+1}\left(\frac{1}{x-y}-\frac{1}{x+1}\right)P(x+1,y+2)\\ & =\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right) \end{align*}

(2)

\begin{align*} \frac{1}{P(x,y)} & =\frac{(x-y)!}{x!}\\ & =\frac{(x-y)!}{x!}\frac{(x)-(x-y+1)}{y-1}\\ & =\frac{1}{y-1}\left(\frac{(x-y)!}{(x-1)!}-\frac{(x-y+1)!}{x!}\right)\\ & =\frac{1}{y-1}\left(\frac{1}{P(x-1,y-1)}-\frac{1}{P(x,y-1)}\right) \end{align*}

(2)-2

\begin{align*} \frac{1}{P(x,y)} & =\frac{1}{x(x-y+1)}\frac{1}{P(x-1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{x-y+1}-\frac{1}{x}\right)\frac{1}{P(x-1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{P(x-1,y-1)}-\frac{1}{P(x,y-1)}\right) \end{align*}

(3)

\begin{align*} Q(x,y) & =\frac{(x+y-1)!}{(x-1)!}\\ & =\frac{(x+y-1)!}{(x-1)!}\frac{(x+y)-(x-1)}{y+1}\\ & =\frac{1}{y+1}\left(\frac{(x+y)!}{(x-1)!}-\frac{(x+y-1)!}{(x-2)!}\right)\\ & =\frac{1}{y+1}\left(Q(x,y+1)-Q(x-1,y+1)\right) \end{align*}

(3)-2

\begin{align*} Q(x,y) & =\frac{1}{(x-1)(x+y)}Q(x-1,y+2)\\ & =\frac{1}{y+1}\left(\frac{1}{x-1}-\frac{1}{x+y}\right)Q(x-1,y+2)\\ & =\frac{1}{y+1}\left(Q(x,y+1)-Q(x-1,y+1)\right) \end{align*}

(4)

\begin{align*} \frac{1}{Q(x,y)} & =\frac{(x-1)!}{(x+y-1)!}\\ & =\frac{(x-1)!}{(x+y-1)!}\frac{(x+y-1)-(x)}{y-1}\\ & =\frac{1}{y-1}\left(\frac{(x-1)!}{(x+y-2)!}-\frac{x!}{(x+y-1)!}\right)\\ & =\frac{1}{y-1}\left(\frac{1}{Q(x,y-1)}-\frac{1}{Q(x+1,y-1)}\right) \end{align*}

(4)-2

\begin{align*} \frac{1}{Q(x,y)} & =\frac{1}{x(x+y-1)}\frac{1}{Q(x+1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{x}-\frac{1}{x+y-1}\right)\frac{1}{Q(x+1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{Q(x,y-1)}-\frac{1}{Q(x+1,y-1)}\right) \end{align*}