Suppose I have probability generating functions $$G_{X}(t) = 0.1t+0.2t^{2}+0.7t^{3}\quad\text{and}\quad G_{Y}(t)=0.5+0.4t^{2}+0.1t^{3}.$$ In other words, the random variable $X$ gets the discrete values $P(X=1)=0.1, P(X=2)=0.2,\ \text{and}\ P(X=3)=0.7$.

How do I calculate the sum or difference of $G_{X}$ and $G_{Y}$? It would feel intuitive to write $$G_{X}(t)+G_{Y}(t) = 0.5 + 0.1t + 0.6t^{2}+0.8t^{3}$$ but the coefficients sum over $1$. If I divide each coefficient by $2$ then they sum again to $1$. Is that the correct to way to approach this?

Suppose I have probability generating functions $$G_{X}(t) = 0.1t+0.2t^{2}+0.7t^{3}\quad\text{and}\quad G_{Y}(t)=0.5+0.4t^{2}+0.1t^{3}.$$ In other words, the random variable $X$ gets the discrete values $P(X=1)=0.1, P(X=2)=0.2,\ \text{and}\ P(X=3)=0.7$.

How do I calculate the sum or difference of $G_{X}$ and $G_{Y}$? It would feel intuitive to write $$G_{X}(t)+G_{Y}(t) = 0.5 + 0.1t + 0.6t^{2}+0.8t^{3}$$ but the coefficients sum over $1$. If I divide each coefficient by $2$ then they sum again to $1$. Is that the correct to way to approach this?

Edit: Sorry for being unclear. I was trying to ask about $G_{X+Y}$, so the PGF of the sum of $X$ and $Y$. The variables can be assumed to be independent.