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. (2.8) j=1 j=1

This is (i). Next, (r ,r ,...,r ) ∈ R(k) ⇒ k r = k and k jr = k ⇒ r +m+ k r = k+m 0 1 k k j=0 j j=0 j 0 j=1 j + + k = ⇒ + ∈ (k+m) and 0(r0 m) j=1 jrj k (r0 m,r1,...,rk) Rk . This proves the necessity of (ii). ∈ (k+m) k = + k = Let (r0,r1,...,rk) Rk be given. Then j=0 rj k m and j=0 jrj k.Then(i) yields that r0 ≥ m, and then r0 −m ∈ N∪{0}. Then we have − ∈ (k) r0 m,r1,...,rk Rk (2.9) − + k = · − + k = because (r0 m) j=1 rj k and 0 (r0 m) j=1 jrj k. This is the sufficiency of (ii). Thus we have proved (ii).

By (ii), the mapping (r0,r1,...,rk) → (r0 + m,r1,...,rk) from every (r0,r1,...,rk) ∈ (k) + ∈ (k+m) Rk to (r0 m,r1,...,rk) Rk is well defined and the mapping is obviously one- to-one, which proves (iii). The proof is completed. ON COMPOSITION OF FORMAL POWER SERIES 765

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 (k) = (k) r0 r1 ··· rk ak Ar a0 a1 ak , r∈ (k) Rk (2.10) (k+m) = (k+m) q0 q1 ··· qk ak Aq a0 a1 ak , q∈ (k+m) Rk the sums have the same number of summands. The number of summands is determined (k) by k only. The number of terms in these two sums are the same, the coefficients, Ar ’s, of terms and the power of a0 may be different.

(k+m) (k) (k) (k+m) To find the relationship between A + and A ,likeAr and Aq (r0 m,r1,...,rk) (r0,r1,...,rk) ∈ N ∪{ } r = ∈ (k) in Corollary 2.3 explicitly, for any k,m 0 and (r0,r1,...,rk) Rk , only apply Lemma 2.2(ii) and (2.7) to the second form in Corollary 2.3. This relationship can be described in the following corollary.

Corollary 2.4. Let k,m ∈ N ∪{0} be given. Let f be a formal power series as in r = ∈ (k) Definition 1.1.IfS is commutative and (r0,r1,...,rk) Rk , then + k+m A(k m) = B(k) , (r0+m,r1,...,r ) (r0,...,r ) k k−r0 k (2.11) k+m + (k+m) = (k) r0 m r1 ··· rk ak Br a0 a1 ak , k−r0 r∈ (k) Rk

(k) where Br is defined as in (2.6). Definition 2.5. ∈ N ∈ N ∪{ } r = ∈ (n) Let n and k 0 be given. If (r0,r1,...,rk) Rk , denote that (n) = r ∈ (n) r = ∀ ≤ ≤ R(s)k Rk r0,0,...,0,rs ,rs+1,...,rk , 1 s k. (2.12)

(n) ⊂ (n) It is obvious that, R(s)k Rk . Lemma 2.6. Let k ∈ N∪{0} and s ∈ N be given and let ks r(s) = ks −k,0,...,0,k,0,...,0 ∈ N∪{0} , (2.13) where the sth coordinate of r(s) is k and the other coordinates of r(s) are zero except the 0’s. Then r ∈ (ks) (i) (s) R(s)ks ; r = ∈ (ks) ≥ − (ii) if (r0,0,...,0,rs ,rs+1,...,rks ) R(s)ks , then r0 k(s 1); (iii) in (ii), r0 = k(s −1)  r = r(s). Proof. r − + = · = r ∈ (ks) For (s), k(s 1) k ks and k s ks imply that (s) Rks .Then(i) follows from Definition 2.5. 766 X.-X. GAN AND N. KNOX

r = ∈ (ks) ⊂ (ks) Next, (r0,0,...,0,rs ,rs+1,...,rks ) R(s)ks Rks implies that

r0 +rs +rs+1 +···+rks = ks, (2.14) srs +(s +1)rs+1 +···+ksrks = ks.

Then, k = rs + ((s +1)/s)rs+1 + ((s +2)/s)rs+2 +···+krks .Sincerj ≥ 0forallj,it follows that

k ≥ rs +rs+1 +···+rks . (2.15) = − ks ≥ − = − Then r0 ks j=s rj ks k k(s 1). This is (ii). Finally, we show (iii). The sufficiency is obvious, we need only show necessity. Sup- pose that r0 = k(s −1). Then,

rs +rs+1 +···+rks = ks −k(s −1) = k, s +1 (2.16) r + r + +···+kr = k. s s s 1 ks

Notice that rj ≥ 0 for all j, s ≤ j ≤ ks. We have

rj = 0,j= s +1,s+2,...,sk. (2.17)

Then rs = ks −r0 = k, and hence r = r(s). The proof is completed.

3. 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

n f(x)= a0 +a1x +···+anx +···, n (3.1) g(x) = b0 +b1x +···+bnx ···, and deg(f ) ≠ 0. Then, the composition g ◦f exists if and only if ∞ n − b an k ∈ S, ∀k ∈ N∪{0}. (3.2) k n 0 n=k

∞ Proof. Suppose that g ◦f exists, that is, c = b a(n) exists for all k, or, the k n=0 n k ∞ (n) series n=0 bnak converges in S for all k. We show that (3.2) is true by mathematical induction on k. Since the conclusion is obvious if a0 = 0, assume that a0 ≠ 0.

It is clear that (3.2) is true for k = 0 because the expression in (3.2)isc0. 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 ), as ≠ 0 but aj = 0 for all 1 ≤ j

Then, ∞ = (n) cks+s bnaks+s n=0 ∞ ks+s−1 = (n) + (n) bnaks+s bnaks+s = + = n ks s  n 0  ∞ (3.6)  n + − +1 r +  =  (ks+s) n r0 (k )s rs ··· ks s  bn Br a0 as aks+s ks +s −r0 n=ks+s r∈ (ks+s) r≠r R(s)ks+s , (s) ∞ ks+s−1 n + − − + b B(ks s)an k 1ak+1 + b a(n) . n k+1 r(s) 0 s n ks+s n=ks+s n=0

r ∈ (ks+s) rs rs+1 ··· rks+s ≠ We may only consider those R(s)ks+s for which as as+1 aks+s 0 in the above r = ∈ (ks+s) expression, and we denote them as (r0,0,...,0,rs ,...,rks+s ) R(s)ks+s .SinceS is a field, we have   ∞ + − + (ks+s) rs rs+1 rks+s  n n r0 (k 1)s  c + = B a a ···a b a ks s r s s+1 ks+s + − n 0 + ks s r0 r ∈ (ks s) r ≠r n=ks+s R(s)ks+s , (s) ∞ ks+s−1 + n − − +B(ks s)ak+1 b an k 1 + b a(n) . r(s) s k+1 n 0 n ks+s n=ks+s n=0 (3.7)

r ≠ r (ks+s) ⇒ + − + ⇒ + − + By Lemma 2.6, (s) in R(s)ks+s r0 >(k 1)s (k 1) ks s r0

∞ k−1 ∞ = (n) = (n) + (n) ck bnak bnak bnak = = = n 0 n 0  n k  (3.11) k−1 ∞  n + − r  = b a(n) + b  B(k) an r0 kar1 ···a k . n k n (r0,...,r ) 0 1 k k−r0 k n=0 n=k ∈ (k) (r0,...,rk) Rk

+ − = − n r0 k = If a0 0, then n>k r0 implies that a0 0, and hence the above series is a finite sum. Then ck exists, and then the conclusion is true. Now we assume that a0 ≠ 0. Since ≠ r ∈ (k) r1 r2 ··· rk ≠ deg(f ) 0, we may only consider those Rk for which a1 a2 ak 0inthe r = above expression, and denote them as (r0,r1,...,rk).Then   k−1 ∞ + − (n)  n (k) n r0 k r1 rk  c = b a + b  B a a ···a  k n k n − (r ,...,r ) 0 1 k k r0 0 k n=0 n=k ∈ (k) (r0,...,rk) Rk   (3.12) k−1 ∞ + − (n) (k) r1 rk  n n r0 k = b a + B a ···a b a n k r 1 k n − 0 k r0 n=0 r ∈ (k) n=k Rk because S is a field. Thus, ck exists in S by (3.2), and we have completed the proof.

Remark 3.2. If a0 = 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. ON COMPOSITION OF FORMAL POWER SERIES 769

Theorem 3.3. Let X be the set of all formal power series from N to the set of complex numbers C.Letf,g∈ X be given with the forms

n n f(x)= a0 +a1x +···+anx +···, g(x) = b0 +b1x +···+bnx +···. (3.13) ∞ n | | ◦ If the series n=0 bnR converges for some R> a0 , then g f exists. Proof. = ≠ If deg(f ) 0, the conclusion is obvious, so assume that deg(f ) 0. ∞ n ∞ n Consider the power series n=0 bnx .Since n=0 bnR converges, it follows that the kth derivative ∞ n−k n(n−1)···(n−k+1)bnx (3.14) n=k converges for |x|

2 Remark 3.4. Let f(x)= a0 +a1x +a2x +···∈X(R) be given. Define the deriva- tive of f to be the formal power series

2 f (x) = a1 +a2x +a3x +···. (3.17)

Then (3.2) is equivalent to (k) g a0 ∈ S, ∀k ∈ N∪{0} (3.18) 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.

Corollary 3.5. Let f,g∈ X(C) be given by

n n f(x)= a0 +a1x +···+anx +···, g(x) = b0 +b1x +···+bnx +···. (3.19)

Suppose that |a0| < 1 and |bn|≤M,foralln ∈ N forsomepositivenumberM.Then g ◦f is well defined. Proof. | | ∞ n Pick R such that a0

Theorems 3.1 and 3.3 tell us that the existence of g ◦ f strongly depends on the constant term of f and the coefficients of g. This result directs us to a deeper investi- gation of the subset of X in which the composition is closed. It is clear that Theorems 3.1 and 3.3 can be applied to ordinary power series.

References

[1] J. Chaumat and A.-M. Chollet, On composite formal power series, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1691–1703. [2] P. Henrici, Applied and Computational Complex Analysis. Vol. 1. Power Series— Integration—Conformal Mapping—Location of Zeros, John Wiley & Sons, New York, 1988. [3] S. Roman, Advanced Linear Algebra, Graduate Texts in Mathematics, vol. 135, Springer- Verlag, New York, 1992.

Xiao-Xiong Gan and Nathaniel Knox: Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA Mathematical Problems in Engineering

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