\[{\operatorname{pq}}^{2}\left(\tfrac{1}{2}z,k\right)=\frac{\operatorname{ps}%
\left(z,k\right)+\operatorname{rs}\left(z,k\right)}{\operatorname{qs}\left(z,k%
\right)+\operatorname{rs}\left(z,k\right)}=\frac{\operatorname{pq}\left(z,k%
\right)+\operatorname{rq}\left(z,k\right)}{1+\operatorname{rq}\left(z,k\right)%
}=\frac{\operatorname{pr}\left(z,k\right)+1}{\operatorname{qr}\left(z,k\right)%
+1}.\]