(1)

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

(2)

\begin{align*} \frac{d}{dy}C(x,y) & =\frac{1}{y!}\frac{d}{dy}P(x,y)+P(x,y)\frac{d}{dy}\frac{1}{\Gamma(y+1)}\\ & =\frac{1}{y!}P(x,y)\psi(1+x-y)+P(x,y)\frac{-\Gamma(y+1)\psi(y+1)}{\Gamma^{2}(y+1)}\\ & =C(x,y)\psi(1+x-y)-C(x,y)\psi(y+1)\\ & =C(x,y)\left\{ \psi(1+x-y)-\psi(1+y)\right\} \\ & =C(x,y)\left\{ H_{x-y}-H_{y}\right\} \end{align*}