If $G_{n+1}=G_n=G$ where $G(s)=G(p+qs)e^{\lambda(s-1)}$, how can one conclude that $G(s)=e^{\lambda(s-1)/p}$
the G's above are generating functions, In this exercise one has independent variables $X_n$ geometric distributed, and writes it as sum of Bernoulli variables, $X_{n+1}=\sum_1^{X_n}B_i+Y_n$ (the factor $e^{\lambda(s-1)}$ comes from the r.v. Y_n. which is Poisson distributed) then every $X_i$ has generating function $p+qs$ and in search of a stationary distribution, one sets $G_{n+1}=G_n=G$, but I don't understand the conclusion.