For an additive function $\lambda$ and an exact sequence of modules
$0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$,
we have $\lambda(M_2) = \lambda(M_1) + \lambda(M_3)$ by definition. If the modules are graded and every morphism preserves degree, then the Poincaré series are also additive, simply by using the additivity of $\lambda$ on every summand of the power series. What if the first morphism merely carries homogeneous elements of degree $d$ to, say, elements of degree $d+k$ for instance? (and similarly for the second morphism) What would the formula for the Poincaré series be in this case?
I figured that I'd simply have to use additivity on a single sequence
$0 \rightarrow M_{1_n} \rightarrow M_{2_{n+deg(f)}} \rightarrow M_{3_{n+deg(f)+deg(g)}} \rightarrow 0$,
but I'm clueless on how to relate the Poincaré series' from here. (I deleted my old unanswered question since it was bloated and hid the main issue I was trying to point out)