We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Prove that, for all integers $n$,
$$P_{n+1}(u,v) = u^{a+1} v^{b} \frac{\partial P_{n}}{\partial u}(u,v) + u^{c} v^{d+1} \frac{\partial P_{n}}{\partial v}(u,v)$$