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On Composition of Formal Power Series IJMMS 30:12 (2002) 761–770 PII. S0161171202107150 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON COMPOSITION OF FORMAL POWER SERIES XIAO-XIONG GAN and NATHANIEL KNOX Received 15 July 2001 2 Given a formal power series g(x) = b0 + b1x + b2x +··· and a nonunit f(x) = a1x + 2 a2x +···, 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. 2000 Mathematics Subject Classification: 13F25, 13J05. 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. 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 Nl to S, where N represents the natural numbers. We denote the set of all such mappings by X(S),orX. 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 = + +···+ n +··· ∞ ⊂ f(x) a0 a1x anx , aj j=0 S. (1.1) In this case, ak, k ∈ N ∪{0} is called the kth coefficient of f .Ifa0 = 0, f is called a nonunit. = ∞ n = Let f and g be formal power series in x with f(x) n=0 fnx and g(x) ∞ n ∈ + · n=0 gnx , and let r S,theng f , rf, and g f are defined as ∞ n (g +f )(x) = g(x)+f(x)= gn +fn x , n=0 ∞ n (r f )(x) = r f (x) = rfn x , (1.2) n=0 ∞ n n (f ·g)(x) = g(x)·f(x)= cnx ,cn = gj fn−j ,n= 0,1,2,.... n=0 j=0 It is clear that all those operations are well defined, that is, g +f , rf, and g ·f are all in X. 762 X.-X. GAN AND N. KNOX We define the composition of formal power series as follows. Definition 1.2. X Let S be a ring with a metric and let be the set of all formal ∈ X = ∞ k power series over S.Letg be given, say g(x) k=0 bkx . We define a subset Xg ⊂ X to be ∞ ∞ X = ∈ X | = k (n) ∈ = g f f(x) akx , bnak S, k 0,1,2,... , (1.3) k=0 n=0 n = ∞ (n) k ∈ N where f (x) k=0 ak x , for all n , created by the product rule in Definition 1.1. We will see that Xg ≠ ∅ by Proposition 1.6. Then the mapping Tg : Xg → X such that ∞ k Tg(f )(x) = ckx , (1.4) k=0 = ∞ (n) = where ck n=0 bnak ,k 0,1,2,..., is well defined. We call Tg(f ) the composition of g and f ; Tg(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. Example 1.3. = R = ∞ n = + Let S .Letg(x) n=0 x and f(x) 1 x. We cannot calculate ∞ n even the first coefficient of the series n=0(f (x)) under Definition 1.2. Thus, the composition g(f(x)) does not exist. Example 1.4. = R = ∞ n = ∞ n Let S , g(x) n=0 x ,andf(x) n=1 n!x . It is clear that ∞ n = the series n=1 n!x 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 se- ries 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. Definition 1.5. = ∞ n Let f(x) n=0 anx be a formal power series. The order of f is the least integer n for which an ≠ 0, and denoted by ord(f ).Thenorm, ,off 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. n Proposition 1.6 (see [3]). Let f(x)= fnx be a formal power series in x.Ifg is a formal power series, such that n lim fng = 0, (1.5) n→∞ ON COMPOSITION OF FORMAL POWER SERIES 763 n then the sum fng converges to a power series. This series is called the composition of f and g and is denoted by f ◦g. n Clearly, the requirement limn→∞ fng =0 implies that the only candidates for such g are formal power series with constant term equal to zero unless f is a poly- nomial. 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 compo- sition of formal power series? Some of these questions are answered in this note. 2. Coefficients of f n(x). A formal power series is actually the sequence of its co- efficients. The composition of formal power series is eventually, or can only be, de- termined 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 coef- ficients can be used to initiate the investigation of the coefficients of f n(x).Weshow n that the kth coefficient of f (x) mainly depends on a0. This property leads to the main theorem. 2 k Definition 2.1. Let f(x)= a0 +a1x +a2x +···+akx +··· be a formal power series. For every n ∈ N,wewrite n = (n) + (n) + (n) 2 +···+ (n) k +··· f (x) a0 a1 x a2 x ak x , (2.1) (1) = ∈ N∪{ } (n) n and put ak ak for all k 0 ; ak is called the kth coefficient of f . If am ≠ 0 but aj = 0forallj>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},thekth (n) coefficient ak is determined by the multinomial 2 k n a0 +a1x +a2x +···+akx (2.2) t (n) only, because at x , t>k,cannot contribute anything to ak . Therefore, by the multi- nomial theorem n n−r n−r −r n−r −r −···−r − (n) = 0 0 1 ··· 0 1 k 1 r0 r1 ··· rk ak a0 a1 ak , r0 r1 r2 rk (2.3) where the sum is taken for all possible nonnegative integers r0,r1,...,rk, such that r0 +r1 +···+rk = n and r1 +2r2 +3r3 +···+krk = k. For any n ∈ N and k ∈ N∪{0},wedenote k k (n) = r | r = ∈ N∪{ } k+1 = = Rk r0,r1,...,rk 0 , rj n, jrj k , (2.4) j=0 j=0 764 X.-X. GAN AND N. KNOX and then define − − − − − −···− (n) n n r0 n r0 r1 n r0 r1 rk−1 Ar = ··· , r0 r1 r2 rk (2.5) (n) ∀r = r0,r1,...,rk ∈ R , k − − − − − −···− (n) n r0 n r0 r1 n r0 r1 rk−1 Br = ··· , r1 r2 rk (2.6) ∀r = ∈ (n) r0,r1,...,rk Rk . (n) = n (n) r ∈ (n) Then Ar r0 Br , for all Rk and (n) = (n) r0 r1 r2 ··· rk ak Ar a0 a1 a2 ak (n) r∈R k (2.7) n = r0 r1 r2 ··· rk Bra0 a1 a2 ak . r0 r∈ (n) Rk ∈ N ∈ N∪{ } k = k = For any n and k 0 ,since j=1 jrj j=0 jrj k for every (r0,r1,...,rk) ∈ (n) Rk , the number of selections of the k-tuple (r1,r2,...,rk) is finite, no matter how large n is. This property, which will be proved in Lemma 2.2, is very important for the (n) investigation of ak . Lemma 2.2. Let k,m ∈ N∪{0} be given. Then ∈ (k+m) ⇒ ≥ (i) (r0,r1,...,rk) Rk r0 m, ∈ (k) + ∈ (k+m) (ii) (r0,r1,...,rk) Rk (r0 m,r1,...,rk) Rk , | (k)|=| (k+m)| (iii) Rk Rk , where |V | denotes the cardinal number of the set V . Proof. ∈ (k+m) ⇒ k = + k = (r0,r1,...,rk) Rk j=0 rj k m and j=0 jrj k. Then, k k r0 = k+m− rj ≥ k+m− jrj = m.
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