where x is a complex variable and A(x) a square matrix of dimension n the entries of which are formal meromorphic power series. Write A = x(A0 +A1x+ · · ·) (A0 6= 0) for the series expansion of A, where the coefficients are matrices over a subfield K of the field of complex numbers. There exists a basis of n formal solutions of the form (see, e.g. Turritin, 1955; Wasow, 1967) yi(t) = etzi(t) (i...

Formal power series are an extension of formal languages. Recognizable formal power series can be captured by the so-called weighted finite automata, generalizing finite state machines. In this paper, motivated by codings of formal languages, we introduce and investigate two types of transformations for formal power series. We characterize when these transformations preserve recognizability, ge...

Formal power series are an extension of formal languages. Recognizable formal power series can be captured by the so-called weighted finite automata, generalizing finite state machines. In this paper, motivated by codings of formal languages, we introduce and investigate two types of transformations for formal power series. We characterize when these transformations preserve rationality, genera...

One of the main virtues of trees is the representation of formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in power series rings. When analyzed in terms of combinatorial Hopf algebras, the simplest examples yield interesting algebraic identities or enumerative...

The class of 2-automatic paperfolding sequences corresponds to the class of ultimately periodic sequences of unfolding instructions. We rst show that a paper-folding sequence is automatic ii it is 2-automatic. Then we provide families of minimal nite-state automata, minimal uniform tag sequences and minimal substitutions describing automatic paperfolding sequences, as well as a family of algebr...

Abstract Let K be a finite field, ( x ) the field of rational functions in over and K formal power series . We show that under certain conditions integral combinations with algebraic coefficients U 1 -number are m -numbers , where is degree extension ), determined by these c...

2 The “algebra” K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 “Algebra” of formal power series under substitution . . . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Substitution of formal power series . . . . . . . . . ....

2 The Riordan skew algebra K[[x]] ⋊ M of formal power series under multiplication and substitution 3 2.1 Basics on formal power series . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Skew algebra of formal power series under substitution . . . . . . . . . . 4 2.2.1 Right-distributive algebras . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Substitution of formal power series . . . . ....

The paper presents the abstract framework of hybrid formal power series. Hybrid formal power series are analogous to non-commutative formal power series. Formal power series are widely used in control systems theory. In particular, theory of formal power series is the main tool for solving the realization problem for linear and bilinear control systems. The theory of hybrid formal power series ...

In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled diierential equations, exponentials of vector elds, and Lie brackets. We show by means of an example |the computation of control variations that give rise to the Legendre-Clebsch condition| how a good choice of formalism , based on expanding diieomorphisms as products of expon...