Is it possible to calculate the result of a nonlinear operator applied to a power series?
In other words, are there closed form solutions for
$$ N(\sum\limits_{n = 0}^\infty a_nx^n) $$whereby $N$ is a nonlinear operator / function?
My only findings so far have been that
$$[\sum\limits_{n = 0}^\infty a_nx^n]^Q, Q \in \mathbb{N}$$
can be explicitly calculated via multiple Cauchy-products.
My only idea so far to handle other nonlinear functions is to use their Taylor-series expansion, i.e. given$$ N(x) = \sum\limits_{n = 0}^\infty q_nx^n $$one could expand$$ N(\sum\limits_{n = 0}^\infty a_nx^n) = \sum\limits_{n = 0}^\infty q_n [\sum\limits_{j = 0}^\infty a_jx^j]^n $$
However, the chance of being able to simplify that is very low, since the general formula for powers of power series is quite ugly (Wikipedia: power of power series)
Any ideas are appreciated, including ways to deal with specific nonlinear operators!
For reference, this question arose from the context of the Homotopy Perturbation Method to solve nonlinear ODEs, since it involves plugging an unknown power series into a nonlinear equation and solving coefficient-wise (equating terms with 0).