On Composition of Formal Power Series

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. 1. Introduction and definitions. It is clear that the concepts of power series and formal power series are related but distinct. So we begin with the definition of formal power series.


Introduction and definitions.
It is clear that the concepts of power series and formal power series are related but distinct.So we begin with the definition of formal power series.Definition 1.1.Let S be a ring, let l ∈ N be given, a formal power series on S is defined to be a mapping from N l to S, where N represents the natural numbers.We denote the set of all such mappings by X(S), or X.
In this note, we only discuss formal power series from N to S. A formal power series f in x from N to S is usually denoted by In this case, a k , k ∈ N ∪ {0} is called the kth coefficient of f .If a 0 = 0, f is called a nonunit.
Let f and g be formal power series in x with f (x) = ∞ n=0 f n x n and g(x) = ∞ n=0 g n x n , and let r ∈ S, then g + f , r f , and g • f are defined as It is clear that all those operations are well defined, that is, g + f , r f , and g • f are all in X.
We define the composition of formal power series as follows.
Definition 1.2.Let S be a ring with a metric and let X be the set of all formal power series over S. Let g ∈ X be given, say g(x) = ∞ k=0 b k x k .We define a subset X g ⊂ X to be k ∈ S, k = 0, 1, 2,... , (1.3) where , for all n ∈ N, created by the product rule in Definition 1.1.We will see that X g ≠ ∅ by Proposition 1.6.Then the mapping T g : X g → X such that k , k = 0, 1, 2,... , is well defined.We call T g (f ) the composition of g and f ; T g (f ) is also denoted by g • f .Some progress has been made toward determining sufficient conditions for the existence of the composition of formal power series.The most recent development can be found in [1] where Chaumat and Chollet investigated the radius of convergence of composed formal power series and obtained some very good results.
Consider the following examples before going any further.
x n and f (x) = 1+x.We cannot calculate even the first coefficient of the series ∞ n=0 (f (x)) n under Definition 1.2.Thus, the composition g(f (x)) does not exist.
converges nowhere except x = 0.However, one checks that the composition g(f (x)), not a composition of functions, is a formal power series.
In Example 1.3 note the difference between the composition of formal power series and the composition of functions such as analytic functions.That is why one is not surprised to read the concern from Henrici [2].Example 1.4 shows that many convergence results in calculus may not be assumed or applied here.Some progress has been made toward determining sufficient conditions for the existence of the composition of formal power series.
x n be a formal power series.The order of f is the least integer n for which a n ≠ 0, and denoted by ord(f ).The norm, , of f is defined as f = 2 − ord(f ) , except that the norm of the zero formal power series is defined to be zero.Under these definitions, a composition was established as follows.
Proposition 1.6 (see [3]).Let f (x) = f n x n be a formal power series in x.If g is a formal power series, such that lim n→∞ f n g n = 0, (1.5) then the sum f n g n converges to a power series.This series is called the composition of f and g and is denoted by f • g.
Clearly, the requirement lim n→∞ f n g n = 0 implies that the only candidates for such g are formal power series with constant term equal to zero unless f is a polynomial.
Is this restriction necessary for the existence of the composition of formal power series?What classes of the formal power series can be allowed to participate in the composition?Additionally, is there any sufficient and necessary condition for composition of formal power series?Some of these questions are answered in this note.

Coefficients of f n (x).
A formal power series is actually the sequence of its coefficients.The composition of formal power series is eventually, or can only be, determined by their coefficients.First, we investigate the coefficients of f n (x) if f (x) is a formal power series.Of course, mathematical induction or the multinomial coefficients can be used to initiate the investigation of the coefficients of f n (x).We show that the kth coefficient of f n (x) mainly depends on a 0 .This property leads to the main theorem.
and put a (1) is called the kth coefficient of f n .If a m ≠ 0 but a j = 0 for all j > m, we define the degree of f to be the number deg(f ) = m.If there is no such a number m, we say that deg(f Suppose that S is commutative and let n ∈ N be given.For any k ∈ N ∪{0}, the kth coefficient a (n) k is determined by the multinomial 3) where the sum is taken for all possible nonnegative integers r 0 ,r 1 ,...,r k , such that For any n ∈ N and k ∈ N ∪{0}, we denote and then define (2.5) (2.6) (2.7) For any n ∈ N and k ∈ N∪{0}, since k , the number of selections of the k-tuple (r 1 ,r 2 ,...,r k ) is finite, no matter how large n is.This property, which will be proved in Lemma 2.2, is very important for the investigation of a . This proves the necessity of (ii).
Let (r 0 ,r 1 ,...,r k ) ∈ R (k+m) k be given.Then k j=0 r j = k+m and k j=0 jr j = k.Then (i) yields that r 0 ≥ m, and then r 0 − m ∈ N ∪{0}.Then we have This is the sufficiency of (ii).Thus we have proved (ii).
By (ii), the mapping (r 0 ,r 1 ,...,r k ) is well defined and the mapping is obviously oneto-one, which proves (iii).
The proof is completed.
Lemma 2.2 gives some significant properties of the coefficients of a formal power series.We use the next corollary to point out these important results.
Corollary 2.3.Let k, m ∈ N ∪ {0} be given.Let f be a formal power series as in Definition 1.1 and let S be a commutative ring.Then, by (2.7), in the expressions (2.10) the sums have the same number of summands.The number of summands is determined by k only.The number of terms in these two sums are the same, the coefficients, r 's, of terms and the power of a 0 may be different.
The proof is completed.

Composition of formal power series.
A formal power series is a mapping from N to a ring S. If this ring is endowed with a metric, the pointwise convergence of a mapping from the set of formal power series to itself is well defined.This gives us a way to define a composition in the set of formal power series over a ring.Theorem 3.1.Let S be a field with a metric, let X be the set of all formal power series from N to S, and let f ,g ∈ X be given with the forms and deg(f ) ≠ 0.Then, the composition g • f exists if and only if exists for all k, or, the series converges in S for all k.We show that (3.2) is true by mathematical induction on k.Since the conclusion is obvious if a 0 = 0, assume that a 0 ≠ 0.
It is clear that (3.2) is true for k = 0 because the expression in (3.2) is c 0 .Suppose that (3.2) holds for all j with 0 ≤ j ≤ k for some k ≥ 0.
Since deg(f ) ≠ 0, we may find s ∈ N such that 1 ≤ s ≤ deg(f ), a s ≠ 0 but a j = 0 for all 1 ≤ j < s, that is, a s is the first nonzero coefficient of f except the constant term.Consider a ( We may only consider those r ∈ R(s) and the right-hand side converges.Notice that S is a field and a s ≠ 0, we have (3.2) for k + 1 and hence we have proved the necessity.Now suppose that (3.9) and the sum is finite.Then, (3.11) If a 0 = 0, then n > k−r 0 implies that a n+r 0 −k 0 = 0, and hence the above series is a finite sum.Then c k exists, and then the conclusion is true.Now we assume that a 0 ≠ 0. Since deg(f ) ≠ 0, we may only consider those r ∈ R because S is a field.Thus, c k exists in S by (3.2), and we have completed the proof.
Remark 3.2.If a 0 = 0, then f is allowed to be in the composition by Theorem 3.1.If g is a polynomial, then (3.2) is true clearly.These results show that Proposition 1.6 is just a special case of Theorem 3.1.Theorem 3.3.Let X be the set of all formal power series from N to the set of complex numbers C. Let f ,g ∈ X be given with the forms Proof.If deg(f ) = 0, the conclusion is obvious, so assume that deg(f ) ≠ 0. Consider the power series converges for all k ∈ N ∪{0}.Then Theorem 3.1 yields the conclusion.
Remark 3.4.Let f (x) = a 0 + a 1 x + a 2 x 2 + ••• ∈ X(R) be given.Define the derivative of f to be the formal power series by the proof of Theorem 3.3.However, Example 1.4 tells us how careful we have to be when we try to assume any result from calculus.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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