In the general model of a Pólya urn ($b = c = 0$, $a = d$), the function $G$ is defined on $U$ by
$$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$
and admits the expansion $G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$ on $D_{\rho}$. The function $H(x,u,v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$ was defined in part III.
Deduce that, for all integers $n$, $P_{n} = Q_{n}$, and then that $H$ and $G$ coincide on $D_{\rho}$.