Associative formal power series in two indeterminates
A formal power series \(F\) in two variables over a commutative field \(\mathbb{K}\) is called associative if it fulfils the associativity equation \[ F(F(X, Y), Z)=F(X, F(Y, Z)). \] The authors deal with the case where \(F(X, Y)\neq X+Y+\dots \) (the case of \(F(X, Y)=X+Y+\dots \) was described by \textit{M. Hazewinkel} [Formal groups and applications. New York-San Francisco-London: Academic Press (1978; Zbl 0454.14020)]). First, they give an example of associative formal power series under consideration. Next, it is shown that the order of an associative formal power series has to be \(1\) or \(2\). Finally, a representation theorem for associative formal power series of order \(2\) (by an invertible formal power series in one variable) is provided, and the convergence of such series in the case where \(\mathbb{K}\in \{\mathbb{R}, \mathbb{C}\}\) is discussed.
order and convergence of associative formal power series
scientific article; zbMATH DE number 6343953
scientific article