On composition of formal power series

Abstract

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

Regular Composition for Slice-Regular Functions of Quaternionic Variable

“…The functions f abs and g abs have power series expansions whose coefficients are real numbers, so all the classical results in [1] (and more recent ones in [3]) which provide sufficient conditions for the existence of the composition of formal power series with coefficients in a commutative field apply. In particular the composition g abs • f abs exists if a 0 = 0.…”

“…. , 3), and i, j , k, are imaginary units (i.e. their square equals −1) such that ij = −ji = k, jk = −kj = i, and ki = −ik = j .…”

“…In this way, H can be considered as a vector space over the real numbers of dimension 4. Given a generic element q = x 0 + ix 1 + jx 2 + kx 3 of H we define in a natural fashion its conjugate q = x 0 − ix 1 − jx 2 − kx 3 , and its square norm |q| 2 = qq = 3 k≥0 x 2 k . The set S = {q ∈ H : q 2 = −1} will be referred to as the sphere of imaginary units of H.…”

Slice regular composition operators

“…Only recently has received some partial answers [3,11] and these results are also extended to the noncommutative setting by Vlacci [26] for slice regular functions. In this article a new slice regular composition of any two slice regular functions will be introduced based on slice regular product.…”

Generating Series for Interconnected Analytic Nonlinear Systems

“…In the process, the set of known properties of the composition product is significantly expanded. (An interesting parallel development has appeared in [3,11] regarding a composition product for formal power series motivated by the composition of two analytic functions (see, e.g., [18]) rather than two Fliess (integral) operators. Its definition is quite distinct and not clearly related to the composition product described in this paper.…”