Abstract
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.
Highlights
Introduction and definitionsIt is clear that the concepts of power series and formal power series are related but distinct
We begin with the definition of formal power series
Some progress has been made toward determining sufficient conditions for the existence of the composition of formal power series
Summary
By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case. We begin with the definition of formal power series. Let S be a ring, let l ∈ N be given, a formal power series on S is defined to be a mapping from Nl to S, where N represents the natural numbers. A formal power series f in x from N to S is usually denoted by f (x) = a0 + a1x + · · · + anxn + · · · , aj In this case, ak, k ∈ N ∪ {0} is called the kth coefficient of f.
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