On composition of formal power series
Xiao-Xiong Gan, Nathaniel Knox · International Journal of Mathematics and Mathematical Sciences · 2002
Given a formal power series g(x) = b0 + b1x + b2x2 + ⋯ and a nonunit f(x) = a1x + a2x2 + ⋯, it is well known that the composition of g with f, g(f(x)), is a formal power series. If the formal power series f above is not a nonunit, that is, the constant term of f is not zero, the existence of the composition g(f(x)) has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series like f above and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.