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On composition of formal power series

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  • Xiao-Xiong Gan
  • Nathaniel Knox

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.

Suggested Citation

  • Xiao-Xiong Gan & Nathaniel Knox, 2002. "On composition of formal power series," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 30, pages 1-10, January.
    • Handle: RePEc:hin:jijmms:197193
    DOI: 10.1155/S0161171202107150
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