A remark on Schröder's equation: Formal and analytic linearization of iterative roots of the power series f(z)=z

in Monatshefte für Mathematik (in press)

We study Schröder’s equation (i.e. the problem of linearization) for local analytic functions F with F (0)=0, F(0)=1, F(0) a root of 1. While Schröder’s equation in this case need not have even a formal ... [more ▼]

We study Schröder’s equation (i.e. the problem of linearization) for local analytic functions F with F (0)=0, F(0)=1, F(0) a root of 1. While Schröder’s equation in this case need not have even a formal solution, we show that if F is formally linearizable, then it can also be linearized by an invertible local analytic transformation. On the other hand, there exist also divergent series solutions of Schröder’s equation in this situation. We give some applications of our results to iterative functional equations, functional-differential equations and iteration groups. [less ▲]

Generalized Dhombres equations in the complex domain a survey

in ESAIM:Proceedings and Surveys (2014, November)

In this survey paper we present the Main Theorems related to the generalized Dhombres equation in the complex domain. We discuss the local as well as the global theory. This includes formal, local ... [more ▼]

In this survey paper we present the Main Theorems related to the generalized Dhombres equation in the complex domain. We discuss the local as well as the global theory. This includes formal, local analytic, entire and meromorphic solutions. [less ▲]

Local solutions of the generalized Dhombres functional equation in a neighbourhood of infinity

in ESAIM: Proceedings and Surveys (2014), 46

On the characterization of generalized Dhombres equations having non constant local analytic or formal solutions

in Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae Sectio Computatorica (2013), 41

We discuss solvability conditions of the generalized Dhombres functional equation in the complex domain. Our focus lies on the so-called 'infinity' case, but also the $z_0$--case is investigated. That ... [more ▼]

We discuss solvability conditions of the generalized Dhombres functional equation in the complex domain. Our focus lies on the so-called 'infinity' case, but also the $z_0$--case is investigated. That means that we consider solutions of a generalized Dhombres equation with initial value $f(\infty)=w_0$, $w_0 \neq 0$, or $f(\infty)=\infty$, or $f(z_0)=1$ for $z_0 \neq 0, \infty$. For both situations we give a characterization of the generalized Dhombres equations which are solvable. [less ▲]

On a functional-differential equation of A. F. Beardon and functional-differential equations of Briot-Bouquet type

in Computational Methods and Function Theory (2013), 13(3), 383-395

We look for solutions of the functional-differential equation f(φ(z))=a(z)f(z)f′(z) for given series φ and a . We show that all formal solutions f of this equation are local analytic. In order to do this ... [more ▼]

We look for solutions of the functional-differential equation f(φ(z))=a(z)f(z)f′(z) for given series φ and a . We show that all formal solutions f of this equation are local analytic. In order to do this we transform the equation to a special type of Briot–Bouquet differential equation. [less ▲]

On the characterization of generalized Dhombres functional equations

Scientific Conference (2013, May 23)

Some remarks to the formal and local theory of the generalized Dhombres functional equation

in Results in Mathematics [=RM] (2013), 63(1-2), 377-395

We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation $f(zf(z))=\varphi(f(z))$ in the complex domain. First we give two proofs of the existence ... [more ▼]

We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation $f(zf(z))=\varphi(f(z))$ in the complex domain. First we give two proofs of the existence theorem about solutions $f$ with $f(0) = w_0$ and $w_0 \in \mathbb{C}^\star \setminus \mathbb{E}$ where $\mathbb{E}$ denotes the group of complex roots of $1$. Afterwards we represent solutions $f$ by means of infinite products where we use on the one hand the canonical convergence of complex analysis, on the other hand we show how solutions converge with respect to the weak topology. In this section we also study solutions where the initial value $z_0$ is different from zero. [less ▲]

Some aspects of the local theory of generalized Dhombres functional equations in the complex domain

in ESAIM: Proceedings and Surveys (2012), 36

We study the generalized Dhombres functional equation f(zf(z)) = ϕ(f(z)) in the complex domain. The function ϕ is given and we are looking for solutions f with f(0) = w0 and w0 is a primitive root of ... [more ▼]

We study the generalized Dhombres functional equation f(zf(z)) = ϕ(f(z)) in the complex domain. The function ϕ is given and we are looking for solutions f with f(0) = w0 and w0 is a primitive root of unity of order l ≥ 2. All formal solutions for this case are described in this work, for the situation where ϕ can be transformed into a function which is linearizable and local analytic in a neighbourhood of zero we also show that we obtain local analytic solutions. We also discuss an example where it is possible to use other methods than we use in the general case. [less ▲]