On composition of formal power series

Xiao-Xiong Gan and Nathaniel Knox

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 30, 1-10

Abstract:

Given a formal power series g ( x ) = b 0 + b 1 x + b 2 x 2 + ⋯ and a nonunit f ( x ) = a 1 x + a 2 x 2 + ⋯ , 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.

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:197193

DOI: 10.1155/S0161171202107150

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