 Solution

The probability generating function encodes the population distribution in one analytic function. Its coefficients and derivative recoverprovided the mean is finite. Higher derivatives at one give factorial moments, and normalization says $\varphi (1,t)=1$.

Interpret the supplied birth and death expressions as transition rates: over a small time $dt$ their probabilities are $(\alpha +\beta n)dt+o(dt)$ and $\gamma ndt+o(dt)$. Incoming probability from $n-1$ and $n+1$, minus both outgoing rates, giveswith $P(-1,t)=0$. Multiplying by $s^n$ and summing yieldsAt stationarity, ${\varphi }_s/\varphi =\alpha /(\gamma -\beta s)$, so integration and normalization giveAlternatively differentiating the evolution equation at one gives $m^{\prime }=\alpha -(\gamma -\beta )m$, so $m(t)=\alpha /(\gamma -\beta )+[m(0)-\alpha /(\gamma -\beta )]e^{-(\gamma -\beta )t}$. If $\beta =0$, the displayed power expression is understood in its limit: $\varphi (s)=e^{(\alpha /\gamma )(s-1)}$, a Poisson distribution with mean $\alpha /\gamma$.