 Solution

Multiply the master equation by $z^n$ and sum over $n$. Shifting indices producesThe apparent negative power causes no problem at $z=0$ when $r_0=0$. Let $D=z{\partial }_z$, so $D{z}^n=n{z}^n$. If the rates are polynomials in $n$, replace their arguments by $D$ to obtain a finite-order partial differential equation:For arbitrary rates the weighted sums still give an exact generating-function evolution, but need not reduce to a finite-order differential operator. The probability generating function obeyswhich is the normalization condition. The evolution has ${\partial }_{t}F(1,t)=0$ when the rate sums converge, so an initially normalized distribution stays normalized. Its derivatives at $z=1$ also give the factorial moments, and expansion around $z=0$ recovers individual probabilities.