The question now is the determination of all associative formal power series of dimension. There are different possibilities to generalize the associativity equation for power series. Namely it can be considered in the sense of formal group laws, or one can discuss it in the meaning of J. Marichal and P. Mathonet see [3].

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Fripertinger, L. Schwaiger, J. Hazewinkel, Formal Groups and Applications , Acad. Press, New York and London, Marichal, P. Mathonet, A description of n-ary semigroups polynomial-derived from integral domains , Semigroup Forum 83 , no. To obtain solutions we have to distinguish different values. If , we obtain an invertible such that the linearization equation holds. If , then there exists an invertible solution if is a Siegel- or Brjuno number see [2] and [1].

Here we ask what can be said about a function if is a root of and is convergent? The computation of solutions which fulfill the generalization is an untouched problem. Furthermore, if we assume that the order of is greater than , what can we say about this generalized equation? Brjuno, Analytic form of differential equations , Trans. Moscow Math. The idea of the Pilgerschritt transform is the following, we demonstrate it for the complex affine group , see [2]:. Let be the above mentioned group with unit element. Let be a -path where. Then we define as the solution of the differential equation.

Then we get a new path by putting.

Then we define the sequence inductively by ,. This sequence is called the sequence of the Pilgerschritt transform of or Pilgerschritt sequence of. Notes Math. Proceedings of the conference, ECIT '87, Netzer, Product-integration and one-parameter subgroups of linear Lie-groups , Iteration theory and its functional equations, Proc.

Recently my attention was drawn to problems occuring in Decision Theory. Here we mainly investigate algebraic properties of polynomials. For an introduction to this topic we refer the reader to the book [1] of my present PostDoc supervisor J. Marichal and his colleagues.

## On functional equations associated with the renormalization of non-commuting circle mappings

Grabisch, J. Marichal, R. Mesiar, E. In the following I want to explain the different areas of my research.

## Si , Cheng : ANALYTIC SOLUTIONS OF A FUNCTIONAL DIFFERENTIAL EQUATION WITH STATE DEPENDENT ARGUMENT

Table of Contents Generalized Dhombres Equations in the complex domain Briot-Bouquet differential equations and functional-differential equations Algebraic aspects of formal power series - associative formal power series Linear equations and non-linear equations Pilgerschritt transform Problems in Decision Theory Generalized Dhombres Equations in the complex domain The generalized Dhombres equation in the complex domain is given by where is a given function and is unknown. For an overview we refer the reader to [3]. Therefore we have functional equations or more generally relations involving our unknown functions and their derived functions.

When we are given one such functional equation as a mathematical model, it is important to try to nd some or all solutions, since they may be used for prediction, estimation and control, or for suggestion of alternate formulation of the original physical model.

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There are many reasons for trying to nd such solutions. Indeed, nding power series solutions are not more complicated than solving recurrence relations or dierence equations. Third, once formal power series solutions are found, we are left with the convergence or stability problem.

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This is a more complicated problem which is not completely solved. F ortunately , there are now several standard techniques which have been proven useful.

In this book, basic tools that can be used to handle power series functions and analytic functions will be given. They are then applied to functional equations in which derived functions such as the derivatives, iterates and compositions of the unknown functions are involved. T o accomplish our ob jective, we keep in mind that this book should be suitable for the senior and rst graduate students as well as anyone who is interested in a quick introduction to the frontier of related research. Only basic second year adv anced engineering mathematics such as the theory of a complex variable and the theory of ordinary dierential equations are required, and a large body of seemingly unrelated knowledge in the literature is presented in an integrated and unied manner.

A synopsis of the contents of the v arious chapters follows.

Some of the material in this book is based on classical theory of analytic functions, and some on theory of functional equations. However, a large number of material is based on recent research works that have been carried out by us and a number of friends and graduate students during the last ten years. Our thanks go to J. Si, X. Wang, T. Lu and J. Lin for their hard works and comments. We tried our best to eliminate any errors. If there are any that have escaped our attention, your comments will be much appreciated. We have also tried our best to rewrite all the material that we draw from v arious sources and cite them in our notes sections.

We beg your pardon if there are still similarities left unattended or if there are any original sources which we have missed.