Exponentiate generating functions as formal power series

Question

In my discrete math class, we are studying generating functions. We learned that $$ e^x = \sum_{i = 0}^{\infty} \frac{x^i}{i!}, $$ which is certainly an identity in calculus. However, in the ring of formal power series, is there any sort of concept of exponentiation? Suppose we have $F, G \in R[[X]].$ Is there any sort of a concept or definition for $$ F^G? $$ I am mostly looking for a reason why the formal power series $e$ (i.e., the power series whose constant coefficient is $e$ and all remaining coefficients are 0), when raised to the power of the power series $x$, satisfies the above identity. A generalization would be appreciated as well, for the $F^G$ case.

Answer 1

Look for the idea of substitution of formal power series. This is called composition in this Wikipedia article. To be more specific with respect to $F^G$, this can be done for example in the case $G=\sum_{i=1}^\infty g_i X^i$ and $F=1+H$ with $H=\sum_{i=1}^\infty h_i X^i$ by considering the formal power series $\exp(G \ln(1+H))$ where $\ln(1+H):=\sum_{i=1}^\infty (-1)^{i-1}\frac{H^i}{i}$.