The Translation Equation in the Ring of Formal Power Series Over ℂ and Formal Functional Equations

Fripertinger, Harald; Reich, Ludwig

doi:10.1007/978-3-319-61732-9_4

Cited by 2 publications

(3 citation statements)

References 15 publications

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“…a holomorphic function A(z) in a neighborhood of the origin, we write A(σ) for the formal series ∞ n=0 a n σ n . While formal power series are a fruitful object of study in their own right [1,3,12,15,19,23,26], mathematicians have a longstanding interest in assigning a definite "sum" to each series, or at least as many series as possible [4,5,6,7,8,13,14,17,18,25,28,29]. For a finitely supported series A, this is straightforward: if A ∈ R[σ], we set A(A) := A(1), and say A(A) is the sum of A.…”

Section: [[Z]] Converges Tomentioning

confidence: 99%

Telescopic, Multiplicative, and Rational Extensions of Summations

Dawson,

Molnar

2021

Preprint

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A summation is a shift-invariant R-module homomorphism from a submodule of R [[σ]] to R or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we revisit telescoping, we study multiplicative closures of summations (such as the usual summation on convergent series) that are not themselves multiplicatively closed, and we study rational extensions as a generalization of telescoping.

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Section: [[Z]] Converges Tomentioning

confidence: 99%

Telescopic, Multiplicative, and Rational Extensions of Summations

Dawson,

Molnar

2021

Preprint

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“…A possible approach for obtaining results concerning the structure of these iteration groups and the explicit form of the coefficient functions is the method of formal functional equations [7].…”

Section: G(x)mentioning

confidence: 99%

“…The method of formal functional equations was so far applied to the construction of iteration groups of formal series and for solving the (associated) system of cocycle equations (see [3,4,5,6,7]). The basic idea of this method is to replace the coefficient functions by certain polynomials in one ore more variables.…”

Section: Introduction: Basic Concepts and Formulation Of The Main Promentioning

confidence: 99%

Families of Commuting Formal Power Series and Formal Functional Equations

Fripertinger

Reich

2020

Annales Mathematicae Silesianae

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In this paper we describe families of commuting invertible formal power series in one indeterminate over 𝔺, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F (x) = σ x + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation.

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The Translation Equation in the Ring of Formal Power Series Over ℂ and Formal Functional Equations

Cited by 2 publications

References 15 publications

Telescopic, Multiplicative, and Rational Extensions of Summations

Telescopic, Multiplicative, and Rational Extensions of Summations

Families of Commuting Formal Power Series and Formal Functional Equations

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