A new approach to the description of one-parameter groups of formal power series in one indeterminate

The aim of the paper is to describe one-parameter groups of formal power series, that is to find a general form of all homomorphisms ΘG:G→Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta_G : G \to \Gamma}$$\end{document} , ΘG(t)=∑k=1∞ck(t)Xk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Theta_G(t) = \sum_{k=1}^{\infty} c_k(t)X^k}$$\end{document} , c1:G→K\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_1 : G \to \mathbb{K} \setminus\{0\}}$$\end{document} , ck:G→K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_k : G \to \mathbb{K}}$$\end{document} for k ≥ 2, from a commutative group (G, + ) into the group (Γ,∘)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\Gamma, \circ)}$$\end{document} of invertible formal power series with coefficients in K∈{R,C}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}}$$\end{document}. Considering one-parameter groups of formal power series and one-parameter groups of truncated formal power series, we give explicit formulas for the coefficient functions ck with more details in the case where either c1 = 1 or c1 takes infinitely many values. Here we give the results much more simply than they were presented in Jabłoński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008). Also the case im c1 = Em (here Em stands for the group of all complex roots of order m of 1), not considered in Jabłoński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008), will be discussed.


Introduction
Let Γ denote the group of all invertible formal power series in one indeterminate with substitution • as a binary operation. By a one-parameter group of formal power series (FPS) we mean any homomorphism Θ G from a group (G, +) into (Γ, •), i.e. any function Θ G : G → Γ satisfying Θ G (t 1 + t 2 )(X) = (Θ G (t 1 ) • Θ G (t 2 ))(X) for t 1 , t 2 ∈ G.
Analogously, we may define a one-parameter group of s-truncated FPS as a homomorphism Θ s G : G → Γ s into the group Γ s of all invertible s-truncated FPS in one indeterminate.
Let F t (X) = F (t, X) := Θ G (t)(X) and F [s] t (X) = F [s] (t, X) := Θ s G (t)(X). If Θ G : G → Γ and Θ s G : G → Γ s are, a one-parameter group of FPS and a one-parameter group of s-truncated FPS, respectively, we will also say that the families (F t (X)) t∈G and (F [s] t (X)) t∈G are a one-parameter group of FPS and a one-parameter group of s-truncated FPS. In this case the families (F t (X)) t∈G and (F [s] t (X)) t∈G satisfy the well known translation equation for t 1 , t 2 ∈ G, F (0, X) = X, (1) in the ring of FPS, and in the ring of s-truncated FPS. Both of them may be written (with F [∞] := F ) in a unified way as for t 1 , t 2 ∈ G, It appears that the form of both one-parameter groups Θ G and Θ s G strongly depend on the function c 1 . We will see that c 1 must be an exponential function, i.e. a homomorphism of (G, +) into (K \ {0}, ·). By the first isomorphism theorem we know that G/ ker c 1 ∼ = im c 1 . There appear the following cases: case 1. c 1 = 1, which means that ker c 1 = G; then Θ G and Θ s G have a rather simple structure; case 2. c 1 = 1, but G/ ker c 1 ∼ = im c 1 is a finite subgroup of (K \ {0}, ·); then im c 1 = E m with some integer m ≥ 2, where E m denotes the set of all roots of 1 of order m, and the general structure of such one-parameter groups is much more complicated; case 3. G/ ker c 1 ∼ = im c 1 is an infinite subgroup of (K\{0}, ·), then the general form of Θ G and Θ s G is also simple.
In order to solve our problem we use algebraic methods jointly with some tools from differential equations and functional equations. We begin with properties of the operation of substitution in the groups Γ and Γ s , which are crucial for our method. Then we find (Theorem 1) regular one-parameter groups of formal power series, i.e. groups with coefficients being C ∞ -functions in the real case and entire when K = C.
Case 1 and case 3 were already investigated in [6,7,9] using connections with the differential groups L 1 s (s ∈ N) and L 1 ∞ . These differential groups will not appear in the present paper. However we will come back to our previous results in cases 1 and 3 since we are now able to give new and simple proofs (see e.g. Theorems 4 and 5). The new idea consists in constructing a particular solution of inhomogeneous functional equations like (33) and (38) using the polynomials obtained in the representations of regular solutions of the translation equation and then adding the simple general solution of the corresponding homogeneous equation, like the Cauchy equation or (35). Our proofs for results such as Theorems 4 and 5 in our previous publications were considerably longer since we had to transfer certain results on differential groups to the groups of invertible formal power series and back.
Case 3 and a subcase of case 2 were already treated in [8] where the standard form of the solution was given whenever it can be applied. Here we give a description of the solutions in this subcase of case 2 using a sequence of polynomials for the representation of the solution (see Corollary 6), but the proof differs in some details (see the functional equation (48)).
Completely new in the present paper is Sect. 10. We give a solution in the following subcase of case 2, namely where we assume that Θ(G) is infinite and the set {lt 0 ∈ G : 0 ≤ l ≤ m − 1} is a subgroup of G and t 0 ∈ G is such that c 1 (t 0 ) is a primitive mth root of 1 (Theorem 6).
As an application of our results we deal with the embedding problems (Sect. 11), which have so far been only studied related to homomorphisms from the group (C, +) into Γ.
At the end of the introduction we would like to mention that our results contain the description of all one-parameter groups Θ : G → Γ s for such groups G as (R, +), (C, +), (R * , ·) and (C * , ·).
It seems that our methods of combining functional equations, differential equations and algebraic methods will not give a satisfactory solution of our problem (i.e. construction of one-parameter groups of FPS) in the remaining subcase. A possible approach to cover also this last open case could come from a theory of families of commuting invertible FPS.
The description of such one-parameter groups uses some sequences of polynomials. Using properties of these sequences (Lemmas 4 and 5), we describe the general form of one-parameter groups in cases 1. and 3 for both FPS and s-truncated FPS. Finally, we consider the second case. In the subcase when one-parameter groups are finite, we give the explicit form of the coefficient 250 W. Jab loński and L. Reich AEM functions of these groups. Also a subcase where c 1 (G)is finite and Θ(G) is infinite is considered.

The ring of formal power series
] will denote the ring of all formal power series (FPS) ∞ k=0 c k X k with coefficients c k ∈ K, where K ∈ {R, C} is the field of real or complex numbers. For a formal power series f (X) = ∞ k=0 c k X k with c i = 0 for some i ∈ N ∪ {0} (N stands here for the set of all positive integers) we define ] : ord f (X) = 1} with the substitution • as a binary operation is a group. Moreover, the set Γ 1 = ∞ k=1 c k X k ∈ Γ : c 1 = 1 is a subgroup of Γ. A very good reference for this topic is [2].
In the sequel we will need the notion of the ring of truncated formal power series. It is known that for a fixed s ∈ N the set We may then define a congruence modulo X s+1 as follows: we say that We consider the quotient ring K , we may associate an s-truncation of a formal power series f (X) defined by In the set K[[X]] s (which may be treated as a set of all polynomials of degree at most s) we introduce, in a natural way, an addition of truncated formal power series. It appears that a multiplication and a substitution must be defined in a specific way so that K[[X]] s should be closed under them. Let and, in the case when ord g(X) ≥ 1, We will also need the notion of a semicanonical form of a formal power series in Γ and Γ s . Namely, for a fixed integer m ≥ 1, let N m be the set of all f (X) ∈ Γ such that f (X) = ∞ k=0 c km+1 X km+1 , whereas N s m stands for the set of all f (X) ∈ Γ s such that f (X) = r k=0 c km+1 X km+1 , where rm + 1 ≤ s < (r + 1)m + 1.
To unify our considerations for both FPS and s-truncated FPS we put Γ ∞ := Γ and N ∞ m := N m . It is known that if ρ ∈ E m is a primitive root of 1 of order m and L ρ (X) = ρX then for both s being a positive integer and s = ∞ we have

Properties of the group operations in Γ s
We give here a crucial formula describing the operation of substitution • in the group Γ ∞ . Because of the construction of Γ s the same formula is also valid in Γ s . Let |k, l| denote the set of all integers n such that k ≤ n ≤ l, and let |k, ∞| be the set of all integers n ≥ k. We assume that t∈∅ a t = 0 and where U n,k := ⎧ ⎨ ⎩ u n := (u 1 , . . . , u n ) ∈ |0, k| n : Note that B un is a multinomial coefficient. As examples of (4) we quote Now, for the convenience of the reader we collect several properties of formulas (4). Although these properties are similar to those of the binary operation in the Lie group L 1 s (cf. [6,Theorem 2]), for the completeness of the paper, we give their proofs here. We mention in the paper, when such a detailed property is needed in a proof, and the reader may postpone the study of these properties until they really appear. Let for an integer p ≥ 0 U p n,k := u n ∈ U n,k : ∀ j∈|2,p+1|,j≤n−1 u j = 0 . Clearly U p2 n,k ⊂ U p1 n,k ⊂ U 0 n,k = U n,k for 0 ≤ p 1 ≤ p 2 . Moreover, we put U 1,1 n,k = U n,k , and, for p ≥ 1, p + 1 = qm with some integers m ≥ 2, q ≥ 1, we define U m,q n,k := u n ∈ U n,k : ∀ j∈|2,qm|∪(|qm+2,n|\Nm) u j = 0 . Then U m,q n,k ⊂ U p n,k . We begin with properties of the sets U n,k , U p n,k and U m,q n,k . (ii) B un = 1 for u n ∈ U n,1 ∪ U n,n ; (iii) if n ≥ 3, k ∈ |2, n − 1| and u n ∈ U n,k , then u j = 0 for every j ∈ |n − k + 2, n| and there exists j ∈ |2, n − k + 1| with u j ≥ 1; (iv) if p ≥ 1 is an integer, n ≥ p + 3 and k ∈ |n − p, n − 1|, then U p n,k = ∅; Proof. The properties (i)-(ii) are simple consequences of the conditions defining the set U n,k .
(iii) If for k ∈ |2, n−1| and u n ∈ U n,k we had u l = 0 for some l ∈ |n−k+2, n|, then, by the conditions defining U n,k we would obtain which leads to a contradiction. Further, suppose that u j = 0 for all j ∈ |2, n − k + 1|. Since also u j = 0 for j ∈ |n − k + 2, n|, then k = n i=1 u i = u 1 = n i=1 iu i = n, which is impossible, so there exists j ∈ |2, n−k +1| with u j ≥ 1. (iv) Let p ≥ 1 be an integer, n ≥ p + 3, k ∈ |n − p, n − 1|. Suppose that U p n,k = ∅. By (iii), for u n ∈ U p n,k there exists then j ∈ |2, n − k + 1| ⊂ |2, p + 1| such that u j ≥ 1, which leads to a contradiction with u n ∈ U p n,k . Hence This contradiction finishes the proof.
When studying one-parameter groups of FPS we will frequently use the following special form of (4).
The following very detailed properties of (4) will be useful in the construction of a one-parameter group of FPS.

One-parameter groups of formal power series
Hence, on account of (5) and Corollary 1 with p = 0, by comparing coefficients, we obtain the infinite system of functional equations 256 W. Jab loński and L. Reich AEM for t 1 , t 2 ∈ G.
In the same way, with for t 1 , t 2 ∈ G. Note that in both cases c 1 must be a generalized exponential function.
Considering at the same time both s ∈ N and s = ∞, we will generally distinguish two cases, c 1 = 1 and c 1 = 1 (in fact, the second one will be divided into subcases later on). In the first case we have clearly the trivial solution c 1 = 1 and c n = 0 for n ∈ |2, s|. So, in the case c 1 = 1, passing over the trivial case, without loss of generality we may assume that there is a nonnegative integer p with p + 2 ≤ s such that c j = 0 for j ∈ |2, p + 1| and c p+2 = 0. Then, by Corollary 1, the systems (11) and (12) may be written as for t 1 , t 2 ∈ G. In the second case we consider the system for t 1 , t 2 ∈ G.

The regular one-parameter groups of formal power series
We describe the regular one-parameter groups of FPS, i.e we give the general regular solution of the systems (13) and (14) for s = ∞. We recall the proofs of these results (see [6,7,9]) not only for the convenience of the reader. We prove these results in a more precise form, with some essential additional consequences, which allow us to give a simple proof of Theorem 2.
In this section we assume that c 1 : or entire functions in the complex case. By the differentiation of each equation of (11) with respect to t 1 , and putting t 1 = 0, jointly with the boundary condition c 1 (0) = 1 and c n (0) = 0 for each n ≥ p+2 (cf. the translation equation (1)), we obtain the system of differential Let us distinguish two cases: In the first case, from the first differential equation of (15) we obtain c 1 = 1. Then, if for some integer p ≥ 0 we have c i (0) = 0 for i ∈ |2, p + 1|, and c p+2 (0) = 0, then from (15) we get c i = 0 for every i ∈ |2, p + 1|, and c p+2 = 0. Hence from (15), for the sequence In the second case, for (λ n ) n≥2 , where λ 1 := c 1 (0) and (k − 1)λ 1 λ k := c k (0) for k ≥ 2, we obtain 258 W. Jab loński and L. Reich AEM There exist sequences of polynomials (L p n ) n≥p+2 and (P n ) n≥2 given by and by such that (i) for every sequence (h n ) n≥p+2 there exists a unique solution of the system of differential equations (15) given by (ii) for every sequence (λ n ) n≥1 with λ 1 = 0, there exists a unique solution of the system of differential equations (16) given by for every j ∈ |2, n − 1|, and let us consider the differential equation with the boundary condition c n (0) = 0. Since the right hand side of (22) is a polynomial in t, this differential equation has a unique solution given by Thus by (18) we can define the sequence of polynomials (L p n ) n≥p+2 such that the solution of (16) is given by (20). Now, let λ 1 = c 1 (0) = 0. Then c 1 (t) = e λ1t and with P 2 (X) = 0. Assume that for some n ∈ N, n ≥ 3 there are polynomials (P j ) j∈|2,n−1 | such that for every j ∈ |2, n − 1|, and let us consider the differential equation 260 W. Jab loński and L. Reich AEM with the boundary condition c n (0) = 0. We have Then, from (23), with c 1 (t) = e λ1t we get The linear differential equation (24) with the boundary condition c n (0) = 0 has exactly one solution (cf. [3, p. 104]) of the form Thus We have thus proved that there is a sequence of polynomials (P n ) n≥2 defined by (19) such that the unique solution of (17) is given by (21). Now we will show that the solutions c k of the systems of differential equations (16) and (17) satisfy the system of equations (11), which means that F (t, X) = ∞ k=1 c k (t)X k with c k given in Theorem 1 are the regular oneparameter groups of formal power series. In the following we use the standard and, in the case when G = K and the coefficient functions c k are differentiable, For G = K the following theorem describes the general regular solution of the translation equation (1) in the ring of FPS. Theorem 2 may be derived from some results in [12,13], but here we give the simple proof based on some ideas from [14].  F (t, X)) t∈K . It is given by the formula H(X) := ∂F ∂t (0, X), in particular, ord H ≥ 1. Proof. First let us assume that (F (t, X)) t∈K is a regular one-parameter group of FPS (i.e. F (t, X) is a solution of the translation equation (1)). By differentiation of (1) with respect to t 1 and putting t 1 = 0 we get (25) with H(X) = ∂F ∂t (0, X), which is nothing else but (15). Conversely, let us consider the system of differential equations (25) (which is equivalent to (16) whenever c 1 (0) = 0, and to (17) if c 1 (0) = 0). We know that for a fixed H(X) ∈ K[[X]] with ord H ≥ 1, the system (25) always has a unique solution. We will show that it is given by (cf. also [12]) where ∂ ∂X denotes the operator of derivation with respect to X. Indeed, it is easy to see that the function F (t, X) given by (26) is a regular solution of the translation equation (1) and F (0, X) = X. Thus, by differentiation of (1) with respect to t 1 and taking  (1). The proof of (iii) is obvious.
As a simple consequence of Theorem 1 and Theorem 2 we obtain Theorem 3. There exist sequences of polynomials (L p n ) n≥p+2 with some integer p ≥ 0, and (P n ) n≥2 , given by (18) and (19), respectively, such that the coefficient functions of every regular nontrivial one-parameter group of FPS F (t, X) =

Properties of the sequences (L n ) and (P n )
Now we collect some properties of the polynomials (L p n ) and (P n ) used for describing regular solutions of (11). First we quote without proof an interesting property of (L p n ) which can be derived from the construction of the regular one-parameter groups of FPS.
with h p+2 = 1. We prove now crucial properties of the polynomials L p n and P n , which allow us to construct the general solution of systems (13) and (14). We begin with a well-known result from polynomial algebra. and in the ring K[X, Y ], where K p p+2 (X) = X, and K p n (X; (h l ) l∈|p+3,n| ) = h n X + L p n (X; (h l ) l∈|p+3,n−p−1| ) for n ≥ p + 3. Proof. By Theorem 3, the family F (t, X) = X + ∞ j=p+2 c j (t)X j with functions c j : K → K for j ∈ |p + 2, ∞| given by c p+2 (t) = t, t ∈ K, c n (t) = h n t + L p n (t; (h l ) l∈|p+3,n−p−1| ), t ∈ K, n ∈ |p + 3, ∞|, 264 W. Jab loński and L. Reich AEM where (h n ) n≥p+3 is an arbitrary sequence of constants (h p+2 = 1), is a regular one-parameter group of FPS with F (t, X) ≡ X mod X 2 . This means that the functions c n for n ≥ p + 2 satisfy the system of equations (13). Then, for each n ≥ p + 3 we have for every t 1 , t 2 ∈ K. This jointly with Lemma 3 implies (29). Moreover, for every n ≥ p + 3 with r = n + p + 1, on account of Corollary 4, we have for t 1 , t 2 ∈ K. Since (K, +) is an abelian group, we get by interchanging t 1 and t 2 (p + 2)t 1 h n t 2 + L p n (t 2 ; (h l ) l∈|p+3,n−p−1| ) + which together with Lemma 3 gives (30).

One-parameter groups in cases 1. and 3.
Let (G, +) be a commutative group, and let s be a positive integer or s = ∞ (by |∞, ∞| we will mean ∅). We know that if F [s] (t, X) = X + s j=2 c j (t)X j is a solution of the translation equation with c 1 = 1, then we find p ≥ 0 such that p + 2 ≤ s, c k = 0 for k ∈ |2, p + 1| and c p+2 : G → K is a nonzero additive function. Then the functions c k : G → K for k ∈ |p + 2, s| satisfy the system of equations (13). We prove Theorem 4. Let the sequence of polynomials (L p n ) n≥p+2 be given by (28) and (18). For every one-parameter group Θ(t)(X) = X + s k=p+2 c k (t)X k of FPS with c p+2 = 0, there exist a nonzero additive function a : G → K, a sequence of additive functions (a n ) n∈|s−p,s| , a n : G → K, and a sequence of constants (h n ) n∈|p+3,s−p−1| such that the coefficient functions (c k ) k∈|p+2,s| are given by 1| )), t ∈ G, n ∈ |s − p, s|. Conversely, for an arbitrary additive function a : G → K, for each sequence (h n ) n∈|p+3,s−p−1| and for every sequence of additive functions (a n ) n∈|s−p,s| , a n : G → K, a function Θ(t)(X) = X + Proof. At first we prove that the functions c n for n ∈ |p + 2, s| given by (32) satisfy the system of equations (13). It is easy to see that the function c p+2 satisfy the first equation of the system (13). So let us fix n ∈ |p + 3, s − p − 1|. Then, using (29), for arbitrary t 1 , t 2 ∈ G we get Finally, fix n ∈ |s − p, s|. Then using (29) with h n = 0 we obtain Consequently, with an arbitrary additive function a n : G → K we get c n (t 1 + t 2 ) = a n (t 1 + t 2 ) + L p n (a(t 1 + t 2 ); (h l ) l∈|p+3,n−p−1| ) = a n (t 1 ) + a n (t 2 ) + L p n (a(t 1 ) + a(t 2 ); (h l ) l∈|p+3,n−p−1| ) = a n (t 1 ) which proves that the functions c n , n ∈ |p+2, s|, defined by (32) satisfy the system of equations (13) with arbitrary constants (h n ) n∈|p+3,s−p−1| and arbitrary additive functions a n : G → K for n ∈ |s − p, s|. Now we are going to show that functions defined by (32) are the only solutions of (13). Here we use an approach much simpler than the one applied in [6,7]. Note that from (13) jointly with c p+2 = 0 it follows that c p+2 is a nonzero additive function. So let 0 = c p+2 = a : G → K be an additive function.
Put h p+2 = 1 and L p p+2 (X) = 0. Assume that for some n ∈ |p Let us consider the equation with a function c n on the left hand side of it, that is for t 1 , t 2 ∈ K, which is, in fact, an inhomogeneous Cauchy equation for c n provided a and (h l ) l∈|p+3,n−p−1| are given. Hence the general solution of (33) is a sum of a particular solution of this equation and an additive function. Moreover, for every t 1 , t 2 ∈ G, on account of (29) with h n = 0, we have which proves that L p n (a(t); (h l ) l∈|p+3,n−p−1| ) is a solution of (33). Thus every solution of (33) must be of the form with an additive function a n : K → K. We will show that a n = h n a for n ∈ |p + 3, s − p − 1|. Let us consider the equation of the system (13) with the function c n+p+1 on the left hand side of it. By (34) and Corollary 4 with r = n + p + 1 it can be written as for t 1 , t 2 ∈ K. Since (G, +) is an abelian group, by interchanging t 1 and t 2 in the above equality we get (p + 2)a(t 1 ) a n (t 2 ) + L p n (a(t 2 ); (h l ) l∈|p+3,n−p−1| ) +na(t 2 ) a n (t 1 ) + L p n (a(t 1 ); (h l ) l∈|p+3,n−p−1| ) From the last relation, using (30) with h n = 0 we get (p + 2)a(t 1 )a n (t 2 ) + na(t 2 )a n (t 1 ) = (p + 2)a(t 2 )a n (t 1 ) + na(t 1 )a n (t 2 ) for t 1 , t 2 ∈ G. Since n ≥ p + 3, we have a(t 1 )a n (t 2 ) = a(t 2 )a n (t 1 ) for t 1 , t 2 ∈ G.
Fix t 1 ∈ G with a(t 1 ) = c p+2 (t 1 ) = 0. Then with h n := an(t1) a(t1) ∈ K we get a n = h n a, and consequently, from (14) we obtain Now, fix n ∈ |s − p, s| (clearly then s < ∞) and assume that Let us consider once more Eq. (33). The same way as before we show that then c n must be of the form (34). Since in this case n + p + 1 > s, this finishes the proof.
Assume now that s is a positive integer or s = ∞ and F [s] (2) in the case when s is finite) with c 1 = 1. Then the functions c i satisfy the system of equations (14) and c 1 is a generalized exponential function. We are going to consider firstly the already mentioned subcase where c 1 takes infinitely many values. In what follows, we will need Lemma 6. Assume that (G, +) is a commutative group, k ≥ 2 is a positive integer and let f : G → K \ {0} be an exponential function such that f (y 0 ) k − f (y 0 ) = 0 for some y 0 ∈ G. If a function g : G → K satisfies the equation where k ≥ 2 is an integer, then there exists a constant p ∈ K such that Proof. From the symmetry of the left hand side of (35), we get for every f (y0) k −f (y0) we obtain (36).

Theorem 5. Let (G, +) be a commutative group which admits a generalized exponential function from
There exists a sequence of polynomials (P n ) n≥2 defined by (19) such that for every oneparameter group Θ(t)(X) = s k=1 c k (t)X k of FPS with a generalized exponential function c 1 taking infinitely many values, there exists a sequence of constants (λ n ) n∈|2,s| such that for every n ∈ |2, s| c n (t) = λ n (c 1 (t) n − c 1 (t)) + c 1 (t)P n c 1 (t); (λ l ) l∈|2,n−1| for t ∈ G. (37) Conversely, for each exponential function c 1 and for each sequence (λ n ) n∈|2,s| , the function Θ(t)(X) = s k=1 c k (t)X k is a one-parameter group of FPS. Proof. The proof of Theorem 5 is based in the same ideas as the proof of Theorem 4 and it is easier than the one given in [9]. We show first that with an arbitrary exponential function c 1 the functions (c n ) n∈|2,s| defined by (37) satisfy the system of equations (14). Indeed, fix n ≥ 2, a sequence (λ n ) n∈|2,s| and t 1 , t 2 ∈ G. Using Lemma 5 we get +c 1 (t 2 ) n λ n (c 1 (t 1 ) n − c 1 (t 1 )) + c 1 (t 1 )P n c 1 (t 1 ); (λ l ) l∈|2,n−1| Now we are going to prove that the functions defined by (37) are the unique ones satisfying the system of equations (14). From (14) it follows that c 1 is an exponential function and assume that it takes infinitely many values. Then, by Lemma 6, where P 2 (X) = 0. Assume now that for some n ∈ |3, s| we have c j (t) = λ n c 1 (t) j − c 1 (t) + c 1 (t)P j c 1 (t); (λ l ) l∈|2,j−1| , t ∈ G, j ∈ |2, n − 1|, and consider the equation of the system (14) with the function c n on the left hand side, i.e.

One-parameter groups in case 2.
We describe the one-parameter groups Θ s : G → Γ s , Θ s (t)(X) = l ≤ m − 1}, mt 0 ∈ ker c 1 and every t ∈ G may be uniquely written as lt 0 + t with some 0 ≤ l ≤ m − 1 and t ∈ ker c 1 . We prove

Proposition 1. Let s be a positive integer or
Conversely, for every one-parameter group of FPS Θ s : ker c 1 → Γ s , Θ s (t)(X) = X + is a one-parameter group of FPS, then, clearly, Θ s | ker c1 = Θ s : ker c 1 → Γ s , Θ s (t)(X) = X + s k=2 c k (t)X k is a one-parameter group of FPS. Put P (X) = Θ s (t 0 )(X). Then mt 0 ∈ ker c 1 , P m (X) = Θ s (t 0 ) m (X) = Θ s (mt 0 )(X) = Θ s (mt 0 )(X), and, since G is a commutative group, for every t ∈ ker c 1 . Finally, for every 0 ≤ l ≤ m − 1, and t ∈ ker c 1 . Now, let us fix a one-parameter group of FPS Θ s : ker c 1 → Γ s 1 , Θ s (t)(X) = X + s k=2 c k (t)X k , and a FPS P (X) = s k=1 d k X k , where d 1 is a primitive root of 1 of the order m, satisfying conditions (39)-(40). Then formula (41) properly defines a function Θ s : G → Γ s . We show that Θ s is then a oneparameter group of FPS. From (39) it follows that for every positive integer n. Then for l 1 t 0 + t 1 , l 2 t 0 + t 2 ∈ G, where 0 ≤ l 1 , l 2 ≤ m − 1 and t 1 , t 2 ∈ ker c 1 , using (42) we get which finishes the proof.
Proposition 1 proved above gives us only a characterization of the solution of the considered problem in the last case. But in some subcases we are able to give also explicit formulas for one-parameter groups of FPS (i.e. explicit solutions of the system of equations (14)). Note that always ker Θ s ⊂ ker c 1 . We give the mentioned formulas in the case when ker Θ s = ker c 1 .
Assume once more that im c 1 = E m with some m ≥ 2. Then, on account of the first isomorphism theorem (cf. [10, p. 16]), we have G/ ker c 1 ∼ = E m , and further, since we assumed ker Θ s = ker c 1 , This means that im Θ s is a finite subgroup of (Γ s , •). From Proposition 1 we deduce then

Corollary 5. Under the assumptions of Proposition
Conversely, for every FPS P (X) = s k=1 b k X k with d 1 being a primitive root of 1 of order m, such that the condition (43) is fulfilled, formula (44) properly defines a function Θ s : G → Γ s , Θ s (t)(X) = s k=1 c k (t)X k and Θ s is a one-parameter group of FPS with im c 1 = E m .
Proof. It is enough to notice only that the finiteness of im Θ s in Proposition 1 implies Θ s (t)(X) = X for t ∈ ker c 1 .
In this case every one-parameter group of FPS Θ s : G → Γ s must be of the form Θ s = Θ s • κ, where κ : G → E m is the canonical homomorphism and Θ s : E m → Γ s , Θ s (z)(X) = s k=1 c k (z)X k is a one-parameter group of FPS which means that functions c 1 : E m → E m with im c 1 = E m , and c k : E m → K satisfy the system of equations Conversely, for each multiplicative function c 1 : E m → E m and for each sequence (λ n ) n∈|2,s| the sequence ( c n ) n∈|2,s| defined by (46) is a solution of the system (45).
Proof. By Theorem 5 we know that c n defined by (46) for n ∈ |2, s| satisfy the system of equations (45). Now, let us consider the second equation of the system (21). Since m ≥ 2, from Lemma 6 we have with P 2 (X) = 0 and some λ 2 ∈ K. Then, analogously as in the proof of Theorem 5, assume that for some n ∈ |3, s| we have for every j ∈ |2, n − 1|, and consider the equation of the system (45) with the function c n on the left hand side, i.e.
The procedure of finding the solution of (47) is almost the same as in Theorem 5 (cf. the solution of Eq. (38)). Note that every solution of Eq. (47) is the sum of a particular solution of this equation (i.e. c 1 (z)P n c 1 (z); (λ l ) l∈|2,n−1| ), and a solution of the equation Let us consider two cases 1. n ≡ 1 mod m, 2. n ≡ 1 mod m.
In the first case, as in the proof of Theorem 5 (then one can easily find z 0 ∈ E m such that c 1 (z 0 ) n − c 1 (z 0 ) = 0) we show that Now we will show that also in the second case the same formula holds, even if it has in some sense a different meaning. Note that for n ≡ 1 mod m, since im c 1 = E m , we have c 1 (z) n = c 1 (z) for z ∈ E m . Thus, from (48) we obtain It is easy to see that if a function f : E m → K satisfies (49), then l : = λ n ( c 1 (z) n − c 1 (z)) + c 1 (t)P n c 1 (t); (λ l ) l∈|2,n−1| , which finishes the proof. Corollary 6. Let (G, +) be a commutative group and let a sequence of polynomials (P n ) n∈|2,s| be given by (19). Every one-parameter group Θ(t)(X) = s k=1 c k (t)X k of FPS such that im c 1 = E m and ker Θ s = ker c 1 is given by (37), where (λ n ) n∈|2,s| is an arbitrary sequence of constants.
Proof. We know that if im c 1 = E m and ker Θ s = ker c 1 then Θ s = Θ s • κ, where κ : G → E m is the canonical homomorphism and Θ s : E m → Γ s , Θ s (z)(X) = s k=1 c k (z)X k is a one-parameter group of FPS. Thus c n = c n • κ for every n ∈ |1, s|. On account of Proposition 2, c n = c n • κ for n ∈ |1, s|, so for every t ∈ G we have = λ n (c 1 (t) n − c 1 (t)) + c 1 (t)P n c 1 (t); (λ l ) l∈|2,n−1| .
This finishes the proof.

Iterative roots of unit in Γ s
Using results of the previous section we describe iterative roots of X in rings of FPS and truncated FPS. Let s be a positive integer or s = ∞. It is easy to see that for a fixed positive integer m ≥ 2, a power series P (X) = s j=1 d j X j is an iterative root of X of order m if and only if Θ Em : E m → Γ s , Θ Em e 2lπi m = P l (X) for l ∈ {0, 1, . . . , m − 1} is a one-parameter group of truncated FPS. Thus the iterative roots of X in the ring of FPS lie in an image of one-parameter groups of formal power series Θ Em (t)(X) = s k=1 c k (t)X k , t ∈ E m . First, note that c 1 = 1. Indeed, if c 1 = 1, then for (G, +) = (E m , ·), every homomorphism a : G → K on a finite group is the zero function (every logarithmic function a : E m → K is the zero function), so (cf. (46)) Θ Em (t)(X) = X for every t ∈ G.
Finally, to describe all iterative roots it is enough to take c 1 = id Em , since in the case c 1 = 1 all functions c k with k ∈ |2, s| are polynomials in c 1 and for every homomorphism c 1 : E m → E m we have im c 1 ⊂ im id Em . In [8] we proved a description of one-parameter groups of FPS using simultaneous conjugation, namely
Next, let h k = 0 for k ∈ |p + 3, s − p − 1| \ N m , a k = 0 for k ∈ |s − p, n − 1| \ N m , with some n ∈ |s − p, s| \ N m . In the same way as above we show that 0 = c n = a n + L p n (a; (h j ) j∈|p+3,n−p−1| ) = a n .