An explicit example of an iteration group in the ring of formal power series

Abstract

We give an example of some iteration group in a ring of formal power series over a field of characteristic 0. It allows us to obtain an explicit formula for some one-parameter group of (truncated) formal power series under an additional condition. Consequently, we are able to show some non-commutative groups of solutions of the third Aczél-Jabotinsky differential equation in the ring of truncated formal power series.

1 Introduction

Let $\varvec{k}$ be a field of characteristic 0 with the prime field $\varvec{q}\subset \varvec{k}$ which is isomorphic to the field $\mathbb {Q}$ of all rational numbers. Assume that $(G,+)$ is a commutative group. For $s\in \mathbb {N}\cup \{\infty \}$ by $\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ we denote the set

$$\begin{aligned} \left\{ \sum _{j=0}^sa_jx^j: a_j\in \varvec{k}\; \text{ for } \,j\in \{0\}\cup \mathbb {N}\,\right\} . \end{aligned}$$

If $s<\infty$ it is the ring of all s-truncated formal power series over $\varvec{k}$. Otherwise $\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_{\infty }$ is the ring of all formal power series over $\varvec{k}$, so we have $\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]=\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_{\infty }$. More details about $\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ are presented in the next section. Let $\Gamma ^s\subset \varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ be the set of all s-truncated formal power series which are invertible with respect to substitution $\circ$ in $\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$. Clearly $(\Gamma ^s,\circ )$ are groups for all $s\in \mathbb {N}\cup \{\infty \}$.

A non-empty family ${\mathcal {F}}=(F_t)_{t\in G}\subset \Gamma ^s$ satisfying

$$\begin{aligned} F_{t_1+t_2}=F_{t_1}\circ F_{t_2}\qquad \text{ for } \,t_1,t_2\in G \end{aligned}$$

is called a one-parameter group of (s-truncated) formal power series. A characterization of one-parameter groups of formal power series can be found among others in [2]. In the case when ${\mathcal {F}}\ni F_t(x)=c_1(t)x+\sum _{j=2}^sc_j(t)x^j$ and either the set $\varvec{F}_1=\{c_1(t)\in \varvec{k}^\star :t\in G\,\}$ is infinite or the family ${\mathcal {F}}=\{F_t:t\in G\,\}$ is finite, one can find $S\in \Gamma ^s$ such that

$$\begin{aligned} F_t(x)=S^{-1}(c_1(t)S(x))\qquad \text{ for } \,t\in G. \end{aligned}$$

The case when $\varvec{F}_1$ is finite but ${\mathcal {F}}$ is infinite is much more complicated and no explicit form of such a group is known. A possible and known description uses sequences of polynomials defined recursively (see [2, 3]).

It was proved in [3, 5, 6] that each element ${\mathcal {F}}\ni \Phi =F_{t_0}$ for $t_0\in G$ of a one-parameter group $(F_t)_{t\in G}$ is a solution of the third Aczél-Jabotinsky formal differential equation

$$\begin{aligned} \frac{{\textrm{ d}}\Phi }{{\textrm{ d}}x}\cdot H=(H\circ \Phi ), \end{aligned}$$

(1)

where $H(x)=\frac{\partial F_t}{\partial t}(x)|_{t=0}$ is the so-called infinitesimal generator of the group $(F_t)_{t\in G}$ (assuming that $(F_t)_{t\in G}$ is formally differentiable). In [3] all groups of solutions of (1) are described in the ring $\varvec{\varvec{k}}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ over an arbitrary field $\varvec{k}$ of characteristic 0. Those descriptions are based on recurrent constructions of two sequences of polynomials over $\varvec{q}$. Earlier results (see [5]) were proved in the ring of formal power series (only the case $s=\infty$) over $\mathbb {C}$. It is known (see [3, 5]) that for $s=\infty$ all possible groups of solutions of (1) are commutative. The situation for finite s is different (cf. [3]) and then also non-commutative groups of solutions appear.

Here we will construct some two-parameter family of formal power series. This will allow us to give explicit forms of groups of solutions of (1) for a specific form of the generator H. In particular cases we obtain also explicit forms of non-commutative groups of solutions of (1).

2 The rings of formal power series and truncated formal power series

In the ring $\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]$ of formal power series $\sum _{j=0}^{\infty }c_jx^j$ over $\varvec{k}$ we define the order of a formal power series by

$$\begin{aligned} {\textrm{ ord}}\,\left( \sum _{j=0}^{\infty }c_jx^j\right) = \min \{j\in \{0\}\cup \mathbb {N}: c_j\ne 0\,\}, \end{aligned}$$

where $\min \emptyset :=\infty$. In the ideal $\mathfrak {m}=(x)=x\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]$ of formal power series f with ${\textrm{ ord}}\,f\ge 1$ we define a substitution in the following way:

$$\begin{aligned} (f\circ g)(x)=\sum _{j=1}^{\infty }c_j\left( \sum _{l=1}^{\infty }d_lx^l\right) ^j \end{aligned}$$

for $f(x)=\sum _{j=1}^{\infty }c_jx^j\in \mathfrak {m}$ and $g(x)=\sum _{j=1}^{\infty }d_jx^j\in \mathfrak {m}$. Then f is invertible with respect to substitution if and only if ${\textrm{ ord}}\,f=1$, whence,

$$\begin{aligned} \Gamma ^{\infty }=\{f\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]:{\textrm{ ord}}\,f=1\,\}. \end{aligned}$$

It is a group under substitution $\circ$ with unit element $L_1(x)=x$.

Let $s\in \mathbb {N}$ be a positive integer. The ring $\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ of s-truncated formal power series is the quotient ring $\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]/\mathfrak {m}^{s+1}$ where

$$\begin{aligned} \mathfrak {m}^{s+1}=x^{s+1}\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]=\{f\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]:{\textrm{ ord}}\,f \ge s+1\,\}. \end{aligned}$$

To each coset $f+\mathfrak {m}^{s+1}$ with $f(x)=\sum _{j=0}^{\infty }c_jx^j\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]$ we associate the s-truncation $f^{[s]}$ of f given by

$$\begin{aligned} f^{[s]}(x):=\sum _{j=0}^sc_jx^j\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s\subset \varvec{k}[{x}]\subset \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]. \end{aligned}$$

In $\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ we introduce operations of addition, multiplication and substitution in the following way:

$$\begin{aligned} \begin{array}{l} (f_1+f_2)(x)=f_1(x)+f_2(x), \\[.5ex] (f_1\cdot f_2)(x)=(f_1\cdot f_2)^{[s]}(x),\\[.5ex] (f_1\circ f_2)(x)=(f_1\circ f_2)^{[s]}(x) \end{array} \end{aligned}$$

for $f_1,f_2\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$. Then $\Gamma ^s$ is the set $\{f\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s:{\textrm{ ord}}\,f=1\,\}$. It is a group under substitution, with unit element $L_1$.

It is known that if $\pi ^k_l:\Gamma ^k\rightarrow \Gamma ^l$ for $k\ge l$ are natural projections defined by l-truncation, then the group $\Gamma ^{\infty }$ can be treated as the projective limit of $(\Gamma ^s)_{s\in \mathbb {N}}$, that is $\Gamma ^{\infty }=\lim _{\leftarrow }\Gamma ^s$ with the canonical projections $\pi ^{\infty }_l:\Gamma ^{\infty }\rightarrow \Gamma ^l$. Moreover, for $s\in \mathbb {N}\cup \{\infty \}$ we put $\Gamma _1^s:=\ker \pi ^s_1$.

For a fixed positive integer n by $\varvec{E}_n\subset \varvec{k}^\star :=\varvec{k}\setminus \{0\}$ we denote the set of all roots of order n of $1\in \varvec{k}$, that is the set of all roots of the polynomial $x^n-1\in \varvec{k}[x]$ in $\varvec{k}$. A root $c\in \varvec{E}_n$ is called primitive of order $n\ge 2$ provided c is not a root of any polynomial $x^k-1$ for $1\le k<n$. By a semicanonical form of order $l\in \mathbb {N}$ in $\Gamma ^s$ we mean any $f(x)=\sum _{j=0}^rc_{jl+1}x^{jl+1}$, where r is either the greatest positive integer with $rl+1\le s$ for finite s, or $r=\infty$. Let ${\mathcal {N}}^s_l$ be the family of all semicanonical forms in $\Gamma ^s$ of order l and let $c\in \varvec{E}_l$ be a primitive root of order l. Put $L_c(x)=cx\in \Gamma ^s$. Then (see [1, Fact 2.2])

$$\begin{aligned} {\mathcal {N}}^s_l=\left\{ f\in \Gamma ^s:f\circ L_c=L_c\circ f\,\right\} , \end{aligned}$$

and thus ${\mathcal {N}}_l^s$ is a subgroup of $\Gamma ^s$. Note that ${\mathcal {N}}^s_1=\Gamma ^s$.

3 Descriptions and properties of the substitution

We will need two descriptions of the substitution law in $\Gamma ^s$. Fix $k,l\in \mathbb {Z}$ with $k\le l$. Put $|k,l|=\{n\in \mathbb {Z}: k\le n\le l\,\}$ and $|k,\infty |=\{n\in \mathbb {Z}:n\ge k\,\}$. We assume that $0^0=1$, $|k,l|=\emptyset$ for $k>l$, $\sum _{t\in \emptyset }a_t=0$ and $\prod _{t\in \emptyset }a_t=1$.

We begin with the following lemma, which is here an important tool in the construction of an iteration group given in the next section.

Lemma 1

(see [4]) Fix $s\in \mathbb {N}\cup \{\infty \}$, $s\ge 2$. If $F_1(x)=\sum _{i=1}^sa_ix^i\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$, $F_2(x)=\sum _{i=1}^sb_i x^i\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$ and $(F_1\circ F_2)(x) =\sum _{n=1}^sd_nx^n\in \varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$, then

$$\begin{aligned} d_n= \sum _{k=1}^n a_k \sum _{\overline{v}_k \in V_{k,n}}\prod _{j=1}^k b_{v_j}\qquad \text{ for } \,n\in |1,s|, \end{aligned}$$

(2)

for every positive integer n, where

$$\begin{aligned} V_{k,n}=\left\{ \overline{v}_k=(v_1,\ldots ,v_k)\in |1,n|^k:\; \sum _{i=1}^k v_i=n \;\right\} \qquad \text{ for } \,1\le k\le n. \end{aligned}$$

For example, for $n=1,2,3$, from (2) we get

$$\begin{aligned} d_1=a_1b_1,\quad d_2=a_1b_2+a_2b_1^2,\quad d_3=a_1b_3+2a_2b_1b_2+a_3b_1^3. \end{aligned}$$

We prove now the characterization of substitution in the subgroup ${\mathcal {N}}^s_l$. For a fixed integer $l\ge 1$ we put ${\mathbb {N}}_l=\{j\in \mathbb {N}: j\equiv 1\,{\textrm{ mod}}\,l\,\}$.

Corollary 1

Fix $r\in \mathbb {N}\cup \{\infty \}$, $l\in \mathbb {N}$. If $F_1(x)=\sum _{j=0}^ra_{jl+1}x^{jl+1}\in \mathcal {N}^{rl+1}_l$ and $F_2(x)=\sum _{j=0}^rb_{jl+1}x^{jl+1}\in \mathcal {N}^{rl+1}_l$, then $(F_1\circ F_2)(x)=\sum _{j=0}^rd_{jl+1}x^{jl+1}x \in \mathcal {N}^{rl+1}_l$ and

$$\begin{aligned} d_{nl+1}=\sum _{k=0}^na_{kl+1}\sum _{ \overline{\nu }_{kl+1}\in \widehat{V}^l_{kl+1,nl+1}}\prod _{j=1}^{kl+1}b_{\nu _jl+1}\; \text{ for } \,n\in |1,r|, \end{aligned}$$

(3)

where

$$\begin{aligned} \widehat{V}_{kl+1,nl+1}^l=\left\{ \overline{\nu }_{kl+1}=(\nu _1,\ldots ,\nu _{kl+1}) \in |0,n-k|^{kl+1}: \sum _{j=1}^{kl+1}\nu _j=n-k\,\right\} \end{aligned}$$

for $1\le k\le n$.

Proof

Since $\mathcal {N}_l^{rl+1}$ is a subgroup of $\Gamma ^{rl+1}$, consequently $(F_1\circ F_2)(x)\in \mathcal {N}_l^{rl+1}$. In order to compute $d_{nl+1}$ for $n\le r$, define

$$\begin{aligned} \widetilde{V}_{kl+1,nl+1}^l=\Bigg \{\overline{v}_{kl+1}= (v_1,\ldots ,v_{kl+1})\in \mathbb {N}_l^{kl+1}:\sum _{i=1}^{kl+1}v_i=nl+1\Bigg \},\; k\in |0,n|. \end{aligned}$$

It is a subset of $V_{kl+1,nl+1}$. We put $a_k=b_k=0$ in (2) for $k\in |2,r|\setminus \mathbb {N}_l$. Since $(F_1\circ F_2)(x)\in \mathcal {N}_l^{rl+1}$, so

$$\begin{aligned} d_{nl+1}=\sum _{k=1}^{nl+1}a_k \sum _{\overline{v}_k \in V_{k,nl+1}}\prod _{j=1}^k b_{v_j}=\sum _{k=0}^na_{kl+1} \sum _{\overline{v}_{kl+1}\in \widetilde{V}_{kl+1,nl+1}}\prod _{j=1}^{kl+1} b_{v_j}, \end{aligned}$$

Furthermore, for $\overline{v}_{kl+1}=(v_1,\ldots ,v_{kl+1})\in \widetilde{V}_{kl+1,nl+1}^l$ we put $v_j=\nu _j l+1\in \mathbb {N}_l$ with $\nu _j\in |0,n|$. Then

$$\begin{aligned} nl+1=\sum _{j=1}^{kl+1}(\nu _jl+1)=l\sum _{j=1}^{kl+1}\nu _j+kl+1, \end{aligned}$$

hence $\sum _{j=1}^{kl+1}\nu _j=n-k$, thus $\nu _j\in |0,n-k|$ for all $j\in |1,kl+1|$. Finally,

$$\begin{aligned} \begin{array}{rcl} d_{nl+1}&{}=&{}\displaystyle \sum _{k=0}^na_{kl+1} \sum _{\overline{v}_{kl+1} \in \widetilde{V}_{kl+1,nl+1}}\prod _{j=1}^{kl+1} b_{v_j}\\ &{}=&{}\displaystyle \sum _{k=0}^na_{kl+1} \sum _{\overline{\nu }_{kl+1} \in \widehat{V}_{kl+1,nl+1}}\prod _{j=1}^{kl+1} b_{\nu _jl+1}. \end{array} \end{aligned}$$

4 The construction

Now, we construct a general example. For fixed $l\ge 1$ and $k\ge 0$ we define the so called l-fold factorial

$$\begin{aligned} (kl+1)!_l:=\prod _{j=0}^k(jl+1), \end{aligned}$$

assuming additionally $(-l+1)!_{l}:=1$. For $l=1$ it coincides with the standard notion of factorial. Moreover, we introduce the following binary operation on $\varvec{k}^{\star }\times \varvec{k}$:

$$\begin{aligned} (y_1,y_2)\diamond (z_1,z_2)=(y_1z_1,y_1z_2+y_2z_1^{l+1}) \qquad \text{ for } \,(y_1,y_2),(z_1,z_2)\in \varvec{k}^{\star }\times \varvec{k}. \end{aligned}$$

Then $(\varvec{k}^\star \times \varvec{k},\diamond )$ is a group isomorphic to $(\widehat{\Gamma }^{l+1},\circ )$, where

$$\begin{aligned} \widehat{\Gamma }^{l+1}:=\{c_1x+c_{l+1}x^{l+1}\in \Gamma ^{l+1}: c_1\in \varvec{k}^\star , c_{l+1}\in \varvec{k}\,\}. \end{aligned}$$

This group is non-commutative and ($\varvec{E}_l\times \varvec{k},\diamond )$ is a commutative subgroup of $(\varvec{k}^\star \times \varvec{k},\diamond )$. Observe that for $l=1$ we have $\widehat{\Gamma }^{2}=\Gamma ^2$ as well as the family

$$\begin{aligned} \widehat{\Gamma }_1^{l+1}:=\{x+c_{l+1}x^{l+1}\in \widehat{\Gamma }^{l+1}: c_{l+1}\in \varvec{k}\,\} \end{aligned}$$

is a commutative group which is isomorphic to $(\{1\}\times \varvec{k},\diamond )\cong (\varvec{k},+)$.

Proposition 1

Fix $r\in \mathbb {N}\cup \{\infty \}$, $l\in \mathbb {N}$. The family $\left( F^{(l)}_{(z_1,z_2)}(x)\right) _{(z_1,z_2) \in \varvec{k}^\star \times \varvec{k}}$,

$$\begin{aligned} F^{(l)}_{(z_1,z_2)}(x)=\sum _{n=0}^r \left( \frac{((n-1)l+1)!_{l}}{n!}\cdot \frac{z_2^n}{z_1^{n-1}}\right) x^{nl+1} \, \text{ for } \,(z_1,z_2)\in \varvec{k}^\star \times \varvec{k}, \end{aligned}$$

(4)

is a non-commutative two-parameter iteration group in ${\mathcal {N}}^{rl+1}_l$ if and only if

$$\begin{aligned} \frac{((n-1)l+1)!_l}{(n-k)!((k-1)l+1)!_{l}}= \sum _{\overline{\nu }_{kl+1}\in \widehat{V}^l_{kl+1,nl+1}} \prod _{j=1}^{kl+1}\frac{((\nu _j-1)l+1)!_{l}}{\nu _j!} \end{aligned}$$

(5)

holds true for all $n\in \mathbb {N}$ and $k\in |0,n|$.

Proof

Fix a positive integer l. We have to show that

$$\begin{aligned} F^{(l)}_{(y_1,y_2)\diamond (z_1,z_2)}=F^{(l)}_{(y_1,y_2)}\circ F^{(l)}_{(z_1,z_2)}\qquad \text{ for } \,(y_1,y_2),(z_1,z_2)\in \varvec{k}^\star \times \varvec{k} \end{aligned}$$

(6)

holds if and only if (5) is satisfied for $n\in \mathbb {N}$ and $k\in |0,n|$. Put

$$\begin{aligned} c_{nl+1}(z_1,z_2)=\frac{((n-1)l+1)!_{l}}{n!}\cdot \frac{z_2^n}{z_1^{n-1}} \quad \text{ for } \,(z_1,z_2)\in \varvec{k}^\star \times \varvec{k},\, n\in \{0\}\cup \mathbb {N}. \end{aligned}$$

On account of Corollary 1 condition (6) is equivalent to

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{n=0}^rc_{nl+1}(y_1z_1,y_1z_2+y_2z_1^{l+1})x^{nl+1}\\[1ex] \displaystyle = \sum _{k=0}^rc_{kl+1}(y_1,y_2)\left( \sum _{j=0}^rc_{jl+1}(z_1,z_2) x^{jl+1}\right) ^{kl+1}\\[1ex] \displaystyle =\sum _{n=0}^r\left( \sum _{k=0}^nc_{kl+1}(y_1,y_2) \sum _{\overline{\nu }_{kl+1}\in \widehat{V}_{kl+1,nl+1}^l}\prod _{j=0}^{kl+1} c_{\nu _{jl+1}}(z_1,z_2)\right) x^{nl+1}\;\,\text{ mod }\hspace{.5mm} x^{rl+2}. \end{array} \end{aligned}$$

We have

$$\begin{aligned} \begin{array}{rcl} c_{nl+1}(y_1z_1,y_1z_2+y_2z_1^{l+1})&{}=&{}\displaystyle \frac{((n-1)l+1)!_{l}}{n!}\frac{(y_1z_2+y_2z_1^{l+1})^n}{(y_1z_1)^{n-1}}\\[1.5ex] &{}=&{}\displaystyle \frac{((n-1)l+1)!_{l}}{n!}\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{\left( y_2z_1^{l+1}\right) ^k\cdot (y_1z_2)^{n-k}}{(y_1z_1)^{n-1}}\\[1.5ex] &{}=&{}\displaystyle \sum _{k=0}^n\frac{((n-1)l+1)!_l}{k!(n-k)!}\frac{y_2^k}{y_1^{k-1}}\cdot \frac{z_2^{n-k}}{z_1^{n-(l+1)k-1}}. \end{array} \end{aligned}$$

Moreover, $\sum _{j=0}^{kl+1}\nu _j=n-k$ for $\overline{\nu }_{kl+1}=(\nu _1,\ldots ,\nu _{kl+1})\in \widehat{V}^l_{kl+1,nl+1}$, hence

$$\begin{aligned}{} & {} \sum _{k=0}^nc_{kl+1}(y_1,y_2)\sum _{ \overline{\nu }_{kl+1}\in \widehat{V}^l_{nl+1,kl+1}}\prod _{j=1}^{kl+1}c_{\nu _{jl+1}} (z_1,z_2)\\{} & {} \quad \displaystyle = \sum _{k=0}^n\frac{((k-1)l+1)!_{l}}{k!}\frac{y_2^k}{y_1^{k-1}} \sum _{\overline{\nu }_{kl+1}\in \widehat{V}^l_{kl+1,nl+1}}\prod _{j=1}^{kl+1} \frac{((\nu _j-1)l+1)!_{l}}{\nu _j!}\frac{z_2^{\nu _j}}{z_1^{\nu _j-1}} \\{} & {} \quad \displaystyle = \sum _{k=0}^n\Bigg (\frac{((k-1)l+1)!_{l}}{k!} \cdot \sum _{\overline{\nu }_{kl+1}\in \widehat{V}^l_{kl+1,nl+1}} \prod _{j=1}^{kl+1}\frac{((\nu _j-1)l+1)!_{l}}{\nu _j!} \Bigg ) \frac{y_2^k}{y_1^{k-1}}\frac{z_2^{n-k}}{z_1^{n-(l+1)k-1}}. \end{aligned}$$

Thus (6) is equivalent to the system (5) for every $n\in \mathbb {N}$ and $k\in |0,n|$. $\square$

Remark 1

Note, that if $l=1$, (5) holds true for every $n\in \mathbb {N}$ and $k\in |0,n|$. It is a consequence of the equality

$$\begin{aligned} \sum _{\overline{\nu }_{k+1}\in \widehat{V}^1_{k+1,n+1}} 1 =\left( {\begin{array}{c}n\\ k\end{array}}\right) \qquad \text{ for } \,n\in \mathbb {N},\,k\in |0,n| \end{aligned}$$

(the number of all compositions of the number $n-k$ into $k+1$ non-negative integers, or, which is the same, the number of all compositions of the number $n+1$ onto $k+1$ positive integers).

Corollary 2

Fix $r\in \mathbb {N}\cup \{\infty \}$, $l\in \mathbb {N}$. If the equalities (5) hold for $n\in \mathbb {N}$ and $k\in |0,n|$, then the iteration group $\left( F^{(l)}_{(z_1,z_2)}(x)\right) _{(z_1,z_2) \in \varvec{k}^\star \times \varvec{k}}$ defined by (4) is isomorphic to $(\widehat{\Gamma }^{l+1},\circ )$.

Proof

Observe that (see the proof of Proposition 1) the coefficient functions $c_{nl+1}$ of the iteration group ${\mathcal {F}}=\left( F^{(l)}_{(z_1,z_2)}(x)\right) _{(z_1,z_2) \in \varvec{k}^\star \times \varvec{k}}$ depend on two variables $(z_1,z_2)\in \varvec{k}^*\times \varvec{k}$. Moreover,

$$\begin{aligned} \pi ^{rl+1}_{l+1}\left( F^{(l)}_{(z_1,z_2)}\right) (x)=z_1x+z_2x^{l+1}\in \widehat{\Gamma }^{l+1} \qquad \text{ for } \,(z_1,z_2)\in \varvec{k}^*\times \varvec{k}. \end{aligned}$$

This implies that the projection $\pi ^{rl+1}_{l+1}|_{\mathcal {F}}$ is injective. Whence $\pi ^{rl+1}_{l+1}:\mathcal {F}\rightarrow \widehat{\Gamma }^{l+1}$ is an isomorphism. $\square$

Since $(\{1\}\times \varvec{k},\diamond )$ is a subgroup of the group $(\varvec{k}^*\times \varvec{k},\diamond )$ and $(\{1\}\times \varvec{k},\diamond )$ is isomorphic to $(\varvec{k},+)$, from Proposition 1 and Corollary 2 one can derive the following result.

Corollary 3

Fix $r\in \mathbb {N}\cup \{\infty \}$ and $l\in \mathbb {N}$. The family $\left( G^{l}_t\right) _{t\in \varvec{k}}$,

$$\begin{aligned} G^{(l)}_t(x)=F^{(l)}_{(1,t)}(x)=\sum _{n=0}^r \left( \frac{((n-1)l+1)!_{l}}{n!}\cdot t^n\right) x^{nl+1}\; \text{ for } \,t\in \varvec{k}, \end{aligned}$$

(7)

is a commutative one-parameter iteration group in ${\mathcal {N}}^{rl+1}_l$ if and only if (5) holds for $n\in \mathbb {N}$ and $k\in |0,n|$. It is isomorphic to $(\varvec{k},+)$.

Remark 2

Fix $r\in \mathbb {N}\cup \{\infty \}$, $l\in \mathbb {N}$ and assume that condition (5) holds for $n\in \mathbb {N}$ and $k\in |0,n|$. For the group $\left( G^{l}_t\right) _{t\in \varvec{k}}$ we have

$$\begin{aligned} \frac{\partial G_t^{(l)}}{\partial t}(x)= \sum _{n=1}^r \left( \frac{((n-1)l+1)!_{l}}{(n-1)!} t^{n-1}\right) x^{nl+1}\; \text{ for } \,t\in \varvec{k}, \end{aligned}$$

hence $H(x)=\frac{\partial G^{(l)}_t}{\partial t}(x)|_{t=0}=x^{l+1}$ is the infinitesimal generator of $\left( G^{(l)}_t\right) _{t\in \varvec{k}}$.

It is known, that (5) is valid for $l=1$ (see Remark 1). We show now that (5) also holds true for an arbitrary positive integer $l\ge 2$ and some values $k\in |0,n|$.

Lemma 2

Condition (5) is trivially satisfied for $k\in \{0,n\}$. Moreover, it is valid for all $n\in \mathbb {N}$ and $k\in |0,n|$, for which $n-k\le 4$.

The proof of the above lemma is very technical and seems to be natural, but we present it for the convenience of the reader.

Proof of Lemma 2

For $k=n$ we have $\widehat{V}^l_{nl+1,nl+1}=\{(0,\ldots ,0)\}$, whereas for $k=0$ we have $\widehat{V}^l_{1,nl+1}=\{(n)\}$. Thus (5) is valid for $k\in \{0,n\}$.

For $k=n-1$ and $\overline{\nu }_{(n-1)l+1}=(\nu _1,\ldots ,\nu _{(n-1)l+1})\in \widehat{V}^l_{(n-1)l+1,nl+1}$ we have $\sum _{j=1}^{(n-1)l+1}\nu _j=n-(n-1)=1$. There are $(n-1)l+1$ sequences with one element equal to 1 and all remaining ones equal to 0. Hence

$$\begin{aligned}{} & {} ((n-2)l+1)!_{l} \sum _{\overline{\nu }_{(n-1)l+1}\in \widehat{V}^l_{(n-1)l+1,nl+1}}\prod _{j=1}^{(n-1)l+1} \frac{((\nu _j-1)l+1)!_{l}}{(\nu _j)!}\\{} & {} \quad \displaystyle =((n-2)l+1)!_l \cdot ((n-1)l+1)\cdot 1= ((n-1)l+1)!_l. \end{aligned}$$

Now, for $k=n-2$ and $\overline{\nu }_{(n-2)l+1}=(\nu _1,\ldots ,\nu _{(n-2)l+1})\in \widehat{V}^l_{(n-2)l+1,nl+1}$ exactly one of the following two possibilities holds:

• (a) either one element of the sequence $\overline{\nu }_{(n-2)l+1}$ is equal to 2 and the remaining ones are equal to 0; there are $(n-2)l+1$ such sequences,

• (b) two elements of the sequence $\overline{\nu }_{(n-2)l+1}$ are equal to 1 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-2)l+1\\ 2\end{array}}\right)$ such sequences.

Thus

$$\begin{aligned}{} & {} ((n-3)l+1)!_{l} \sum _{\overline{\nu }_{(n-2)l+1}\in \widehat{V}^l_{(n-2)l+1,nl+1}}\prod _{j=1}^{(n-2)l+1} \frac{((\nu _j-1)l+1)!_{l}}{(\nu _j)!}\\{} & {} \displaystyle =((n-3)l+1)!_l\cdot \left( ((n-2)l+1)\cdot \frac{l+1}{2} +\left( {\begin{array}{c}(n-2)l+1\\ 2\end{array}}\right) \cdot 1\right) \\{} & {} \displaystyle = \frac{((n-1)l+1)!_l}{2!}. \end{aligned}$$

For $k=n-3$ and $\overline{\nu }_{(n-3)l+1}=(\nu _1,\ldots ,\nu _{(n-3)l+1})\in \widehat{V}^l_{(n-3)l+1,nl+1}$ exactly one of the following possibilities holds:

• (a) one element of the sequence $\overline{\nu }_{(n-3)l+1}$ is equal to 3 and the remaining ones are equal to 0; there are $(n-3)l+1$ such sequences,

• (b) one element of the sequence $\overline{\nu }_{(n-3)l+1}$ is equal to 2, another one is equal to 1 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-3)l+1\\ 2\end{array}}\right) \cdot 2$ such sequences,

• (c) three elements of the sequence $\overline{\nu }_{(n-3)l+1}$ are equal to 1 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-3)l+1\\ 3\end{array}}\right)$ such sequences.

Then

$$\begin{aligned}{} & {} ((n-4)l+1)!_{l} \sum _{\overline{\nu }_{(n-3)l+1}\in \widehat{V}^l_{(n-3)l+1,nl+1}}\prod _{j=1}^{(n-3)l+1} \frac{((\nu _j-1)l+1)!_{l}}{(\nu _j)!}\\{} & {} \displaystyle =((n-4)l+1)!_l\cdot \left( ((n-3)l+1)\cdot \frac{(l+1)(2l+1)}{3!} \right. \\{} & {} \displaystyle \qquad \left. +\,2\cdot \left( {\begin{array}{c}(n-3)l+1\\ 2\end{array}}\right) \cdot \frac{l+1}{2}+\left( {\begin{array}{c}(n-3)l+1\\ 3\end{array}}\right) \cdot 1\right) =\frac{((n-1)l+1)!_l}{3!}. \end{aligned}$$

Finally, for $k=n-4$, $\overline{\nu }_{(n-4)l+1}=(\nu _1,\ldots ,\nu _{(n-4)l+1})\in \widehat{V}^l_{(n-4)l+1,nl+1}$ exactly one of the following possibilities holds:

• (a) one element of the sequence $\overline{\nu }_{(n-4)l+1}$ is equal to 4 and the remaining ones are equal to 0; there are $(n-4)l+1$ such sequences,

• (b) one element of the sequence $\overline{\nu }_{(n-4)l+1}$ is equal to 3, another one is equal to 1 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-4)l+1\\ 2\end{array}}\right) \cdot 2$ such sequences,

• (c) two elements of the sequence $\overline{\nu }_{(n-4)l+1}$ are equal to 2 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-4)l+1\\ 2\end{array}}\right)$ such sequences,

• (d) one element of the sequence $\overline{\nu }_{(n-4)l+1}$ is equal to 2, two others are equal to 1 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-4)l+1\\ 3\end{array}}\right) \cdot 3$ such sequences,

• (e) four elements of the sequence $\overline{\nu }_{(n-4)l+1}$ are equal to 1 and the remaining ones are equal to 0; there are $\left( {\begin{array}{c}(n-4)l+1\\ 4\end{array}}\right)$ such sequences.

Then

$$\begin{aligned}{} & {} ((n-5)l+1)!_{l} \sum _{\overline{\nu }_{(n-4)l+1}\in \widehat{V}^l_{(n-4)l+1,nl+1}}\prod _{j=1}^{(n-4)l+1} \frac{((\nu _j-1)l+1)!_{l}}{(\nu _j)!}\\{} & {} \displaystyle =((n-5)l+1)!_l\cdot \left( ((n-4)l+1)\cdot \frac{(l+1)(2l+1)(3l+1)}{4!} \right. \\{} & {} \displaystyle \quad +\,2\cdot \left( {\begin{array}{c}(n-4)l+1\\ 2\end{array}}\right) \cdot \frac{(l+1)(2l+1)}{3!}+\left( {\begin{array}{c}(n-4)l+1\\ 2\end{array}}\right) \cdot \left( \frac{l+1}{2}\right) ^2\\{} & {} \displaystyle \quad \left. +\,3\cdot \left( {\begin{array}{c}(n-4)l+1\\ 3\end{array}}\right) \cdot \frac{l+1}{2}+\left( {\begin{array}{c}(n-4)l+1\\ 4\end{array}}\right) \cdot 1\right) =\frac{((n-1)l+1)!_l}{4!}. \end{aligned}$$

This completes the proof. $\square$

Remark 3

Observe that on account of Lemma 2 condition (5) holds true for $n\le 5$ and $k\in |0,n|$.

Since (5) is always satisfied with $l=1$, we obtain what follows.

Corollary 4

Fix $s\in \mathbb {N}\cup \{\infty \}$ with $s\ge 2$. Then

$$\begin{aligned} F^{(1)}_{(z_1,z_2)}(x)=z_1 x+z_2x^2+\sum _{n=2}^s\frac{z_2^n}{z_1^{n-1}}x^{n+1}\qquad \text{ for } \,(z_1,z_2)\in \varvec{k}^\star \times \varvec{k}, \end{aligned}$$

is a non-commutative two-parameter iteration group of invertible formal power series. It is an injective embedding of the group $\Gamma ^2$ into $\Gamma ^s$. In particular,

$$\begin{aligned} G^{(1)}_t(x)=F^{(1)}_{(1,t)}(x)=x+tx^2+\sum _{n=2}^st^nx^{n+1}\qquad \text{ for } \,t\in \varvec{k}, \end{aligned}$$

is a commutative one-parameter iteration group of formal power series over $\varvec{k}$ with infinitesimal generator $H(x)=\frac{\partial G^{(1)}_t}{\partial t}(x)|_{t=0}=x^2$.

The group $\left( F^{(1)}_{(z_1,z_2)}\right) _{(z_1,z_2)\in \varvec{k}^*\times \varvec{k}}$ is isomorphic to $\Gamma ^2$, whereas $\left( G_t^{(1)}\right) _{t\in \varvec{k}}$ is isomorphic to $(\varvec{k},+)$.

In order to describe solutions of some special case of the Aczél-Jabotinsky differential equation we need the following groups (see [3]). For $l,s\in \mathbb {N}$ with and $2l+1\le s$ let us consider the product $\varvec{E}_l\times \varvec{k}^{l+1}$ with an operation $\overline{\diamond }:(\varvec{E}_l\times \varvec{k}^{l+1}) \times (\varvec{E}_l\times \varvec{k}^{l+1})\rightarrow \varvec{E}_{l+1} \times \varvec{k}^{l+1}$,

$$\begin{aligned}{} & {} (c_1,(c_j)_{j\in \{l\}\cup |s-l+1,s|})\overline{\diamond } (d_1,(d_j)_{j\in \{l\} \cup |s-l+1,s|})\\&\qquad&\displaystyle =(c_1d_1,(c_1d_j+d_1^jc_j)_{j\in \{l\}\cup |s-l+1,s|}) \end{aligned}$$

for $(c_1,(c_j)_{j\in \{l\}\cup |s-l+1,s|}),(d_1,(d_j)_{j\in \{l\}\cup |s-l+1,s|}) \in \varvec{E}_l\times \varvec{k}^{l+1}$. Similarly, if $l,s\in \mathbb {N}$ with $l+1\le s\le 2l$, we use the product $\varvec{E}_l\times \varvec{k}^l$ with an operation $\widehat{\diamond }:(\varvec{E}_l\times \varvec{k}^l) \times (\varvec{E}_l\times \varvec{k}^l)\rightarrow \varvec{E}_l\times \varvec{k}^l$ defined by

$$\begin{aligned} (c_1,(c_j)_{j\in |s-l+1,s|})\widehat{\diamond }(d_1,(d_j)_{j\in |s-l+1,s|}) =(c_1d_1,(c_1d_j+d_1^jc_j)_{j\in |s-l+1,s|}) \end{aligned}$$

for $(c_1,(c_j)_{j\in |s-l+1,s|}),(d_1,(d_j)_{j\in |s-l+1,s|}) \in \varvec{E}_l\times \varvec{k}^l$. Observe that for $l\ge 2$ the groups $(\varvec{E}_l\times \varvec{k}^{l+1},\overline{\diamond })$ and $(\varvec{E}_l\times \varvec{k}^l,\widehat{\diamond })$ are not commutative provided $\{1\}\subsetneq \varvec{E}_l$.

From [3] one can derive the following result.

Lemma 3

[3, Corollaries 5 and 6] Fix $r\in \mathbb {N}\cup \{\infty \}$ and a positive integer l. Assume that $(G_t)_{t\in \varvec{k}}$, $G_t(x)=x+tx^{l+1}+\sum _{j=2}^rc_{jl+1}(t)x^{jl+1}\in {\mathcal {N}}^{rl+1}_l$ with some $c_{jl+1}:\varvec{k}\rightarrow \varvec{k}$ for $j\in |2,r|$, is a one-parameter group of solutions of the differential equation

$$\begin{aligned} \frac{{\textrm{ d}}\Phi }{{\textrm{ d}}x}\cdot x^{l+1}=\left( \Phi (x)\right) ^{l+1} \end{aligned}$$

(8)

in the ring $\varvec{k}[\hspace{-0.05cm}[{x}]\hspace{-0.05cm}]_s$, where either $rl+1\le s<(r+1)l+1$ for finite r or $s=\infty$ otherwise.

• (i) For $s=\infty$ the family $(\widetilde{G}_{d,t})_{(d,t)\in \varvec{E}_l\times \varvec{k}}$,

$$\begin{aligned} \widetilde{G}_{d,t}=dx+tx^{l+1}+\sum _{j=2}^{\infty } dc_{jl+1}(d^{-1}t)x^{jl+1} \end{aligned}$$

is the group of all solutions of (8). It is isomorphic to $(\varvec{E}_l\times \varvec{k},\diamond )$.

• (ii) For $s\in |2l+1,\infty |$ the family $\left( \widetilde{G}_{d_1,t,d_{s-l+1},\ldots ,d_s} \right) _{(d_1,t,d_{s-l+1},\ldots ,d_s)\in \varvec{E}_l\times \varvec{k}^{l+1}}$ defined by

$$\begin{aligned} \begin{array}{rcl} \widetilde{G}_{d_1,t,d_{s-l+1},\ldots ,d_s}(x)&{}=&{} \displaystyle d_1x+tx^{l+1}+ \sum _{j=2}^{r-1}d_1c_{jl+1}(d_1^{-1}t)x^{jl+1}+\sum _{j=s-l+1}^{rl} d_jx^j \\[1ex] &{} &{} \displaystyle \qquad +(d_1c_{rl+1}(d_1^{-1}t)+d_{rl+1})x^{rl+1}+ \sum _{j=rl+2}^s d_jx^j, \end{array} \end{aligned}$$

is the group of all solutions of (8). It is isomorphic to $(\varvec{E}_l\times \varvec{k}^{l+1},\overline{\diamond })$.

• (iii) For $2\le l+1\le s\le 2l$ the family $\left( \widetilde{G}_{d_1,d_{s-l+1},\ldots ,d_s} \right) _{(d_1,d_{s-l+1},\ldots ,d_s)\in \varvec{E}_l\times \varvec{k}^l}$ defined by

$$\begin{aligned} \widetilde{G}_{d_1,d_{s-l+1},\ldots ,d_s}(x)=d_1x+ \sum _{j=s-l+1}^s d_jx^j \end{aligned}$$

is the group of all solutions of (8). It is isomorphic to $(\varvec{E}_l\times \varvec{k}^l,\widehat{\diamond })$.

Applying Corollary 4 we give an explicit form of the group of all solutions of the third Aczél-Jabotinsky formal differential equation (AJ) in the case $H(x)=x^2$, that is

$$\begin{aligned} \frac{{\textrm{ d}}\Phi }{{\textrm{ d}}x}\cdot x^2=\left( \Phi (x)\right) ^2. \end{aligned}$$

(9)

Putting $l=1$, thus $d=1$, we obtain:

Corollary 5

• (i) The family $(G^{(1)}_t)_{t\in \varvec{k}}$,

$$\begin{aligned} G^{(1)}_t(x)=x+tx^2+\sum _{n=2}^{\infty }t^nx^{n+1} \qquad \text{ for } \,t\in \varvec{k}, \end{aligned}$$

is the group of all solutions of (9) for $s=\infty$. It is isomorphic to $(\varvec{k},+)$ and so commutative.

• (ii) The family $(\widehat{G}^{(1)}_{(t,c)})_{(t,c)\in \varvec{k}^2}$,

$$\begin{aligned} \widehat{G}^{(1)}_{(t,c)}(x)=x+tx^2+\sum _{n=2}^{s-2}t^nx^{n+1}+ (c+t^{s-1})x^s \qquad \text{ for } \,(t,c)\in \varvec{k}^2, \end{aligned}$$

is the group of all solutions of (9) for $s\in \mathbb {N}$, $s\ge 3$. It is isomorphic to $(\varvec{k}^2,+)$ and so commutative.

• (iii) The family $\widehat{\Gamma }^2_1=\{x+tx^2:t\in \varvec{k}\,\}$ is the group of all solutions of (9) for $s=2$. It is isomorphic to $(\varvec{k},+)$ and so commutative.

According to Corollary 3, similar results for solutions of the formal differential equation (8) can be proved under the assumption that (5) holds true.

Corollary 6

Fix an integer $l\ge 2$ and assume that (5) holds for $n\in \mathbb {N}\cup \{0\}$ and $k\in |0,n|$.

• (i) The family $(G^{(l)}_{(d,t)})_{(d,t)\in \varvec{E}_l\times \varvec{k}}$

$$\begin{aligned} G^{(l)}_{(d,t)}(x)= & {} \sum _{n=0}^{\infty } \left( \frac{((n-1)l+1)!_{l}}{n!}\cdot \frac{t^n}{d^{n-1}}\right) x^{nl+1}\\= & {} dx+tx^{l+1}+\sum _{n=2}^{\infty } \left( \frac{((n-1)l+1)!_{l}}{n!}\cdot \frac{t^n}{d^{n-1}}\right) x^{nl+1} \text{ for } \,(d,t)\in \varvec{E}_l\times \varvec{k}, \end{aligned}$$

is the group of all solutions of (8) for $r=\infty$. It is isomorphic to $(\varvec{E}_l\times \varvec{k},\diamond )$ and so commutative.

• (ii) The family $(\widehat{G}^{(l)}_{(d_1,t,d_{s-l+1},\ldots ,d_s)})_{(d_1,t,d_{s-l+1}, \ldots ,d_s) \in \varvec{E}_l\times \varvec{k}^{l+1}}$,

$$\begin{aligned}{} & {} {\widehat{G}^{(l)}_{(d_1,t,d_{s-l+1},\ldots ,d_s)}(x)= \displaystyle d_1x+tx^{l+1}+ \sum _{n=2}^{r-1}\frac{t^n}{d_1^{n-1}}x^{nl+1}+\sum _{j=s-l+1}^{rl} d_jx^j+}\\{} & {} \displaystyle \!\!\!\left( d_{rl+1}+\frac{t^{rl+1}}{d_1^{rl}}\right) x^{rl+1}+ \sum _{j=rl+2}^sd_jx^j \text{ for } (d_1,t,d_{s-l+1},\ldots ,d_s)\in \varvec{E}_l\times \varvec{k}^{l+1}, \end{aligned}$$

is the group of all solutions of (8) for a finite integer $s\ge 2l+1$, where $r\in \mathbb {N}$ is such that $rl+1\le s< (r+1)l+1$. It is isomorphic to $(\varvec{E}_l\times \varvec{k}^{l+1},\overline{\diamond })$ and so non-commutative provided $\{1\}\subsetneq \varvec{E}_l$.

• (iii) The family $\{cx+c_{s-l+1}x^{s-l+1}+\ldots +c_sx^s:c\in \varvec{E}_l, c_{s-l+1},\ldots ,c_s\in \varvec{k}\,\}$ is the group of all solutions of (8) for $s\in |l+1,2l|$, which is isomorphic to $(\varvec{E}_l\times \varvec{k}^l,\widehat{\diamond })$ and so non-commutative provided $\{1\}\subsetneq \varvec{E}_l$.

Remark 4

We know that (5) holds true for all $n\in \mathbb {N}$ and $k\in |0,n|$. Since the proof of this fact uses a completely new approach, it will be proved in a separate paper.