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 ...