On composition of formal power series

• Main Authors: Xiao-Xiong Gan, Nathaniel Knox
• Format: Article
• Language: English
• Published: Hindawi Limited 2002-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/S0161171202107150

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.