International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 12, Pages 761-770

On composition of formal power series

Xiao-Xiong Gan and Nathaniel Knox

Department of Mathematics, Morgan State University, Baltimore 21251, MD, USA

Received 15 July 2001

Copyright © 2002 Xiao-Xiong Gan and Nathaniel Knox. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.