= Solution
The <probability generating function> encodes the population distribution in one analytic function. Its coefficients and <derivative> recover
$$
P(n,t)=\frac1{n!}\partial_s^n\phi(0,t),\qquad\langle n\rangle=\partial_s\phi(1,t),
$$
provided the mean is finite. Higher <derivatives> at one give <factorial moments>, and normalization says $\phi(1,t)=1$.
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 $\gamma n\,dt+o(dt)$. Incoming <probability> from $n-1$ and $n+1$, minus both outgoing rates, gives
$$
\partial_tP(n,t)=(\alpha+\beta(n-1))P(n-1,t)+\gamma(n+1)P(n+1,t)-[\alpha+(\beta+\gamma)n]P(n,t),
$$
with $P(-1,t)=0$. Multiplying by $s^n$ and summing yields
$$
\boxed{\phi_t=\alpha(s-1)\phi+(s-1)(\beta s-\gamma)\phi_s.}
$$
At stationarity, $\phi_s/\phi=\alpha/(\gamma-\beta s)$, so <integration> and normalization give
$$
\boxed{\phi(s)=\left(\frac{\gamma-\beta}{\gamma-\beta s}\right)^{\alpha/\beta},\qquad\langle n\rangle=\frac{\alpha}{\gamma-\beta}.}
$$
Alternatively differentiating the evolution equation at one gives $m'=\alpha-(\gamma-\beta)m$, 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: $\phi(s)=e^{(\alpha/\gamma)(s-1)}$, a <Poisson distribution> with mean $\alpha/\gamma$.
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