Power series in one variable

Scheinberg, Stephen

doi:10.1016/0022-247x(70)90028-4

Cited by 33 publications

(30 citation statements)

References 10 publications

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“…In order to present these relations we recall the basic definitions and properties of analytic iteration groups of local analytic functions and their description by Acz e el-Jabotinsky differential equations (see [8], [9], [10], also [15]). A family ðF t Þ t 2 C of formal power series F t ðxÞ ¼ c 1 ðtÞx þ Á Á Á with c 1 ðtÞ 6 ¼ 0 ðt 2 CÞ is an analytic iteration group, if all coefficient functions t 7 !…”

Section: Generalized Dhombres Equations and Iteration Groupsmentioning

confidence: 99%

Local Analytic Solutions of the Generalized Dhombres Functional Equation I

Reich¹

2007

sam

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We study local analytic solutions f of the generalized Dhombres equation f ðx f ðxÞÞ ¼ 'ð f ðxÞÞ with f ð0Þ ¼ 0 in the complex domain. We give an existence result, describe the structure of the set of all local analytic solutions and solve the converse problem, i.e., we characterize those local analytic functions which are solutions of a generalized Dhombres equation. Connections of generalized Dhombres equations with linear functional equations and generalized Böttcher equations are used. Furthermore, we establish relations of generalized Dhombres equations with Briot-Bouquet differential equations and with iteration groups. Finally, as an application of Böttcher functions, we describe the connections between two generalized Dhombres equations and the representations of their solutions as infinite products. (2000): Primary 30D05, 34M25, 39B12, 39B32; Secondary 30B10. Mathematics Subject Classification

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Section: Generalized Dhombres Equations and Iteration Groupsmentioning

confidence: 99%

Local Analytic Solutions of the Generalized Dhombres Functional Equation I

Reich¹

2007

sam

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“…We note that Lie-Gröbner-series in the context of iteration groups have already been used by St. Scheinberg [22] and also by L. Reich and J. Schwaiger in [21].…”

Section: P N (C K (S) + C K (T)) = P N (C K (S + T)) = C N (S + T) = mentioning

confidence: 99%

Formal functional equations and generalized Lie–Gröbner series

Fripertinger

Reich

2017

Journal of Difference Equations and Applications

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Studying the translation equation F(s + t, x) = F (s, F(t, x)), s, t ∈ C, for F t (x) = F(t, x) = ν≥1 c ν (t)x ν , t ∈ C, or the associated system of cocycle equations in rings of formal power series it is well known that the coefficient functions of their solutions are polynomials in additive and generalized exponential functions. Replacing these functions by indeterminates we obtain formal functional equations. Applying formal differentiation operators to these formal equations we obtain different types of formal differential equations. They can be solved in order to get explicit representations of the coefficient functions. In the present paper we consider iteration groups of type II, i.e. solutions of the translation equation of the form where H is the formal generator of G. Using this particular form of the formal iteration group G we are able to find short proofs and elegant representations of the solutions of the cocycle equations. In connection with the second cocycle equation we study the generalized Lie-Gröbner operator

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“…We note that Lie-Gröbner-series in the context of iteration groups have already been used by St. Scheinberg [11] and also in [9].…”

Section: Consider Two Integer Sequencesmentioning

confidence: 99%