Formal functional equations and generalized Lie–Gröbner series

Studying the translation equation , , for , , or the associated system of cocycle equations in rings of formal power series it is well known that the coefficient functions of their solutions are polynomials in additive and generalized exponential functions. Replacing these functions by indeterminates we obtain formal functional equations. Applying formal differentiation operators to these formal equations we obtain different types of formal differential equations. They can be solved in order to get explicit representations of the coefficient functions. In the present paper we consider iteration groups of type II, i.e. solutions of the translation equation of the form , , where and is an additive function different from 0. They correspond to formal iteration groups of type II, which turn out to be the Lie–Gröbner series . Here the Lie–Gröbner operator D is defined by for where H is the formal generator of G. Using this particular form of the formal iteration group G we are able to find short proofs and elegant representations of the solutions of the cocycle equations. In connection with the second cocycle equation we study the generalized Lie–Gröbner operator , , where are given. It yields the corresponding generalized Lie–Gröbner series which appears in the presentation of the solution of the second cocycle equation.


Introduction
In [4] we have introduced the method of 'formal functional equations' to solve the translation equation (and the associated system of cocycle equations) in rings of formal power series over C in the case of iteration groups of type I. In [5][6][7] we applied this method also for the translation equation and the associated cocycle equations in rings of formal power series over C in the case of iteration groups of type II. Formal functional equations in connection with the translation equation were also studied by D. Gronau [10,11]. Now we will show that for iteration groups of type II the method of Lie-Gröbner series allows to present some elegant proofs.
Let C [[x]] be the ring of formal power series f (x) = ν≥0 c ν x ν over C in the indeterminate x. For a detailed introduction to formal power series we refer the reader to [1] and [16]. Together with addition + and multiplication · the set C[[x]] forms a commutative ring. If f = 0, then the order of f is defined as ord (f ) = min{n ≥ 0 | c n = 0}. Moreover, ord (0) = ∞. The composition • of formal series is defined as follows: ] ν . This series converges with respect to the order topology in C [[x]]. Consider then ( , •) is the group of all invertible formal power series (with respect to •).
We consider the translation equation for F t (x) = F(t, x) = ν≥1 c ν (t)x ν ∈ , t ∈ C. (Cf. the introduction of [4] for the motivation to study (T) and basic results on its solutions (F t ) t∈C .) A family (F t ) t∈C which satisfies (T) is called iteration group, and neglecting the trivial iteration group, there are two types of such groups, namely iteration groups of type I where the coefficient c 1 is a generalized exponential function different from 1, and iteration groups of type II, where c 1 = 1.
It is known that for each iteration group of type II there exists an integer k ≥ 2 such that where c k : C → C is an additive function different from 0. A family (F t ) t∈C is an iteration group of type II, if and only if the system is satisfied for all s, t ∈ C, whereP n are universal polynomials which are linear in c k (s), . . . , c n−k (s). Comparing coefficients in c ν (s + t) = c ν (t + s), ν ≥ 2k, we can prove that there exists a sequence of polynomials (P n ) n≥k so that According to (1) these polynomials must satisfy P n (c k (s) + c k (t)) = P n (c k (s + t)) = c n (s + t) = P n (c k (s)) + P n (c k (t)) + kc k (s)P n−(k−1) (c k (t)) +P n c k (s), . . . , P n−k (c k (s)), c k (t), . . . , P n−k (c k (t)) , for all s, t ∈ C and n ≥ k, where P j = 0 for j < k andP j = 0 for j ≤ 2k.
Since the image of c k contains infinitely many elements we can prove for any polynomial Q(x, y) ∈ C[x, y] that Q(c k (s), c k (t)) = 0 for all s, t ∈ C implies Q = 0. From (2) we obtain by replacing c k (s) and c k (t) by independent variables y, z, that P n (y + z) = P n (y) + P n (z) + kyP n−(k−1) (z) + n − (k − 1) P n−(k−1) (y)z +P n y, . . . , P n−k (y), z, . . . , P n−k (z) (3) for all n ≥ k.
We call G(y, x) a formal iteration group of type II. It also satisfies the condition Iteration groups of type II and formal iteration groups of this type are related in the following way. Theorem 1: F(s, x) = x + c k (s)x k + n>k P n (c k (s))x n is a solution of (T) if and only if G(y, x) = x + yx k + n>k P n (y)x n is a solution of (Tform) and (B).
For formal series f (x), H(x) ∈ C[[x]] consider the differential operator Iterative powers of D are defined as All operators D n are linear, thus for f 1 n ≥ 0, and moreover D satisfies the product rule and more general For a detailed introduction to Lie-Gröbner series see [8] or [9, chapter 1]. The operator LG is linear and multiplicative which means that LG and LG Then ).
Since f is the limit of f k with respect to the order topology we obtain the Commutation Theorem (cf. [ We note that Lie-Gröbner-series in the context of iteration groups have already been used by St. Scheinberg [22] and also by L. Reich and J. Schwaiger in [21].

Formal iteration groups of type II are Lie-Gröbner series
In order to determine all formal iteration groups of type II we are looking for relations between the solutions G(y, x) of (Tform) and the formal generator H(x) of G defined by Here h k = 1. (Notice that in the situation of an analytic iteration group the coefficient of Differentiation of (Tform) with respect to z together with the mixed chain rule and putting z = 0 yields Since the solutions of (Tform) are elements of (C[y]) [[x]] it is possible to write them in the form This allows us to rewrite (PDform) and (B) as holds true for all n ≥ 0. By induction we derive from (5 n ) that where is the Lie-Gröbner operator. Theorem 3: (1) If G is a solution of (Tform) and (B), then it is a solution of (PDform), whence it is the Lie-Gröbner series LG y (x) where H is the formal generator of G. (2) For any generator H(x) = x k + n>k h n x n , k ≥ 2, the unique solution G(y, x) = LG y (x) of (5) and (6) is a solution of (Tform) and (B).
Proof: The first assertion is proved above. Now we show that G(y, x) = LG y (x) is a formal iteration group of type II: In other words the composition of two Lie-Gröbner series is again a Lie-Gröbner series.
LG y (LG z (x)) = LG y+z (x) Let G(y, x) be a formal iteration group of type II with formal generator H(x). Since G(y, x) = LG y (x) we get as an immediate consequence of the Commutation Theorem (Theorem 2) the Commutation Theorem for iteration groups of type II. Theorem 4: Let G(y, x) be a formal iteration group of type II. Then for any power series K(x) of order at least 1 we have G(y, K(x)) = K(G(y, x)). (8)

Remark 5:
Let H be the formal generator of the formal iteration group G of type II. Since Moreover Thus

Remark 6:
The general idea of formal functional equations and Lie-Gröbner series is the following: We start with a functional equation like the translation equation or the cocycle equations (introduced later). From these equations we determine formal equations by replacing independent values by independent variables. In order to solve these formal equations we derive by purely algebraic differentiation and by applying mixed chain rules some differential equations, which we are able to solve. After reordering the summands of a solution we derive a representation as a (generalized) Lie-Gröbner series. Finally we prove that this (generalized) Lie-Gröbner series is a solution of the formal equation we wanted to solve. This idea will be applied to the first and second cocycle equations. In connection with the problem of a covariant embedding of the linear functional equation ϕ(p(x)) = a(x)ϕ(x) + b(x) with respect to an iteration group (F(t, x)) t∈C (cf. [2,3]) we have to solve the two cocycle equations under the boundary conditions These cocycle equations appear also in other settings, see e.g. [12,13,15,20], or [14].

The first cocycle equation
We study the first cocycle equation for under the boundary condition where (F t ) t∈C is an iteration group of type II. Then α 0 is a generalized exponential function andα(t, x) := α(t,x) α 0 (t) is also a solution of (Co1). By substitution into the logarithmic series we obtain that γ (t, x) := log (α(t, x)) = n≥1 γ n (t)x n is a solution of satisfying γ (0, x) = 0, if and onlyα(t, x) is a solution of (Co1) and (B1).
By comparing coefficients it is easy to prove Lemma 7: Let F(t, x) = x + n≥k P n (c k (t))x n be an iteration group of type II, then each coefficient function γ n (t) of a solution γ of (Co1log) is a polynomialP n (c k (t)), t ∈ C. Moreover for all s, t ∈ C we have n≥1P Replacing c k (s) and c k (t) by independent variables y, z, we obtain the formal first cocycle where G(y, x) = x + yx k + n>k P n (y)x n is a formal iteration group of type II. As a consequence we easily obtain Theorem 8: Let c k = 0 be an additive function. Then γ (s, x) = n≥1P n (c k (s))x n is a solution of (Co1log) satisfying γ (0, x) = 0 if and only if (y, x) = n≥1P n (y)x n is a solution of (Co1form) satisfying (B1 ).
Differentiation of (Co1form) with respect to z together with the mixed chain rule and putting z = 0 yields where K(x) := ∂ ∂y (y, x)| y=0 and H(x) is the formal generator of the formal iteration group G(y, x). We call K the generator of the solution of (Co1form).
Thus ∂ ∂y (y, This allows us to rewrite (Co1PD) as (9) is satisfied if and only if holds true for all n ≥ 2. By induction we prove that the unique solution of (9) with ((B1 )) is where D : . This is a generalization of a Lie-Gröbner series. Given LG ξ (K(x)) dξ.

Theorem 9:
(1) If is a solution of (Co1form) and (B1 ) with given generator K, then it is a solution of (Co1PD). Thus it has a representation of the form (10) where D(f ) = f H and H is the formal generator of the iteration group G. (2) Let G be a formal iteration group of type II with formal generator H. For any series K(x) of order at least 1 the unique solution (y, x) of (Co1PD) and (B1 ) is a solution of (Co1form).

Proof:
The first assertion is clear. The proof of the second assertion is based on the Commutation Theorem for iteration groups of type II (Theorem 4).
Let x, y, z be distinct indeterminates.
where α 0 is a generalized exponential function, , c k an additive function and G(y, x) a formal iteration group of type II. Moreover, each solution of (Co1) and (B1) is of this form.
Since the order of K 2 is at least k, H is a divisor of K 2 , thus there existsK 2 Consequently, We note in passing that each of the three factors of α(s, x) is a solution of (Co1) and (B1), thus they are units in C [[x]]. Moreover P(y, x) and which follows from (Co1form) and (B1 ). LetP By construction the coefficients of x n in P(y, x) andP(y, x) are polynomials in y.

The second cocycle equation
is a formal iteration group of type II, c k = 0 an additive function, and α a solution of (Co1) and (B1), then we study the second cocycle equation for Using the particular form of α given in Theorem 10 we study and Obviously, (s, x) depends on the non-trivial additive function c k and on the generalized exponential function α 0 . We write (s, x) = n≥0 n (s)x n .
In [7] we distinguish four cases which cover all possible choices of P(y, x).
In each case we determine a formal equation from (Co2 ) by replacing independent values c k (s), c k (t), s, t ∈ C by indeterminates U and V . If α 0 = 1, then the independent values c k (s), c k (t), α 0 (s), α 0 (t), s, t ∈ C are replaced by indeterminates U, V , S and T (cf. [4,Lemma 16.6]). If the additionally occurring additive function A is not a scalar multiple of c k , then A and c k are linearly independent, and according to [7,Lemma 2] the values c k (s), c k (t), A(s), A(t), s, t ∈ C can be replaced by four indeterminates U, V , σ , τ . These four formal equations are combined in which we want to solve under the boundary condition where Let H be the generator of the formal iteration group G of type II of order k, and consider we have from (PDform), (B) and (11) that , x)). If we consider the generalized Lie-Gröbner operator If κ 1 = · · · = κ k−1 = 0, then D = D. The next two technical lemmata are easy to prove. Lemma 12: For f 1 U, x)) .
We also mention the following rule for the generalized Lie-Gröbner operator D: Proof: By induction we prove the assertion together with the claim that the series ]. For n = 0 the assertions are obvious. Assume that the assertions hold true for n ≥ 0.

Let
LG ( LG Proof: We prove that both and (14) has a unique solution. Simple computations show that and (0, x) = f (x). By Lemma 13 we have and by induction we obtain and R is uniquely determined by (14). The uniqueness would also follow from (the formal part of) a uniqueness theorem for parameter dependent differential equations in the complex domain. LG( and LG U (LG V (f )) = LG U+V (f ).
Proof: The first assertion is trivial. Concerning the second we apply Theorem 15 obtaining what follows from (Co1form ) forP and (Tform) for G.

An immediate consequence of Lemma 13 and Theorem 15 is
LG U (f ) .
In general, the generalized Lie-Gröbner operator LG is not multiplicative. To be more precise: Conversely, if κ 1 = · · · = κ k−1 = 0, then the generalized operator LG coincides with the Lie-Gröbner operator LG which is multiplicative.
The next Lemma describes a relation between a series f and its image D(f ). Its proof is straight forward.
and the assertion follows sinceP(U, x) = 0. Conversely, assume that ∂ ∂U LG U (f (x)) = 0, then and consequently D j (f ) = 0 for all j ≥ 1. Consequently the first four assertions are equivalent. If In order to get detailed information on the coefficient functions of a solution of (Co2form) we deduce the following differential equations from the second cocycle equation. By differentiating (Co2form) with respect to S (U and σ ) and setting S = 1, U = 0, and σ = 0 we get where are the three generators of R.
We consider a solution R(S, Uσ , x) of (Co2form) as an element of (C First we study the situation λ = 0. (1) If R is a solution of (Co2form) and (B2 ), then R satisfies the three equations (Co2PD1)-(Co2PD3), and is of the form This is a generalization of a generalized Lie-Gröbner series. Moreover the generators must satisfy the conditions N 1 = 0 and D(N 3 (x)) = 0. (2) If N 1 = 0, D(N 3 (x)) = 0, then the system consisting of (Co2PD1)-(Co2PD3), (B2 ) has a unique solution, which is the substitution of N 2 into a primitive of a generalized Lie-Gröbner series. Moreover this solution satisfies (Co2form).

Proof:
If R satisfies (Co2form) and ( B2 ) then also the three differential equations. From Since the left hand side is a multiple of S whereas the right hand side does not depend on S it follows that x)).
Now we prove the second assertion. If R(S, U, σ , x) is a solution of the three formal differential equations and the boundary condition, then from N 3 = 0 and (Co2PD3) we get R(S, U, σ , x) =R(S, U, x). Due to λ = 1 and (Co2PD1) it follows thatR(S, U, x) = −N 1 (x) + S n≥0R n (x)U n , whereR n (x), n ≥ 0, still must be determined. From  R(1, 0, 0 Comparing the coefficients of S 0 we derive that N 2 (x) = D (N 1 (x)). Moreover, by induction we obtain thatR n (x) = 1 n! D n (N 1 (x)) for n ≥ 0. Therefore, the solution R(S, U, σ , x) is uniquely determined as SinceP(U, x) satisfies (Co1form ), G satisfies (Tform) and according to Theorem 15 we derive In a similar way by differentiation of (Co2form) with respect to T (V and τ ) and substituting T = 1, V = 0, and τ = 0 we obtain another system of differential equations, namely where are the three generators of R.
Working with a method different from the application of Lie-Gröbner series we proved the following two theorems in [7] describing the solutions of this system together with the boundary condition (B2 ). It is possible to apply the method of Lie-Gröbner series also for this problem. Comparing these proofs we see that especially in the situation λ = 0 the method of Lie-Gröbner series allows more elegant and simpler proofs. Theorem 23: Let λ = 0.
(1) If R is a solution of (Co2form) and (B2 ), then R satisfies the three Equations (Co2D1)-(Co2D3), and it has a representation as a generalized Lie-Gröbner series of the form LG ξ (N 2 (x)) dξ.
In the second and third case assume that f (x) = n≥0 f n x n , then D(f (x)) = n≥0 f n D(x n ). Moreover for n ≥ 0 we have thus ord (D(x n )) = r + n and consequently f = 0 is the unique solution of D(f ) = 0.
In the third case κ k−1 = n 1 ∈ N, thus ord (D(x n )) = k − 1 + n only for n = n 1 , and ord (D(x n 1 )) > k − 1 + n 1 . Comparing coefficients of x n in D(f ) = 0 we obtain that f n = 0 for n < n 1 , the coefficient f n 1 is not determined by this equation, actually it can be chosen arbitrarily in C, and the coefficients f n , n > n 1 , are uniquely determined depending on f n 1 ∈ C.  According to the proof of Theorem 25 in the second case we have ord (D(f )) ≥ r. If ord (N) ≥ r, then f is uniquely determined by D(f ) = N.
In the third case we obtain from the proof of Theorem 25 that ord (D(f )) ≥ k − 1. If ord (N) ≥ k − 1, then comparison of coefficients (or an application of the Theory of Briot-Bouquet equations, cf. [ Finally we want to give the explicit form of the solutions β of (Co2). According to (12) we have , α 0 is a generalized exponential function, c k = 0 is an additive function, G(y, x) is a formal iteration group of type II, κ 1 , . . . , k k−1 ∈ C, P(y, x) = exp A(s), x), where R is a solution of (Co2form), and A is another additive function so that c k and A are linearly independent.
Writing N 2 (x) as k−2 j=0 n j x j +Ñ 2 (x) where ord (Ñ 2 ) ≥ k − 1, by Theorem 26 there exist a seriesN 2 ∈ C[[x]] and a constant b ∈ C so that D(N 2 ) + bx k+n 1 −1 =Ñ 2 , and we conclude that These results generalize the representations of β given in Theorems 16, 17 and 18 of [3] or the second and third item of Theorem 2.8 of [2]. Combining the two systems of differential equations for the formal second cocycle R we obtain the following system of three formal Aczél-Jabotinsky equations where and are the three generators of R. Only (Co2AJ2) is a differential equation for R.
If conversely R satisfies the three equations where N 3 = 0 and D(N 1 (x)) = N 2 (x), then R is of the form R(S, U, σ , x) = SLG U (N 1 (x)) − N 1 (x) and according to Theorem 22 it is a solution of (Co2form) and (B2 ).
In the situation λ = 0 we do not obtain the same solutions as for the two other systems of formal equations. Now (Co2AJ1) and (Co2AJ3) read as N j (x) = LG (N j (x) LG ξ (N 2 (x)) dξ +R(S, U, σ , x) where D(R(S, U, σ , x)) = 0. Again by Theorem 25 if all the coefficients κ j , 1 ≤ j ≤ k −1, of P(U, x) are equal to zero, thenR(S, U, σ , x) can be any element of C[S, U, σ ] since it must be constant with respect to x. Taking still into account that N 1 (x) = c 1 ∈ C, N 2 (x), N 3 (x) = c 3 ∈ C are the generators of R the polynomialR still has to satisfy ∂ ∂SR (S, 0, 0, x)| S=1 = c 1 , LG ξ (N 2 (x)) dξ is a solution of the three formal Aczél-Jabotinsky equations, the boundary condition (B2 ), and the three conditions imposed by the generators of R but not of (Co2form). Also in the situation κ 1 = · · · = κ k−2 = 0 and κ k−1 = n 1 ∈ N there exist solutions of the three formal Aczél-Jabotinsky equations which are not solutions of (Co2form).