Displaying similar documents to “Remainders of power series.”

Further remarks on formal power series

Marcin Borkowski, Piotr Maćkowiak (2012)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we present a considerable simplification of the proof of a theorem by Gan and Knox, stating a sufficient and necessary condition for existence of a composition of two formal power series. Then, we consider the behavior of such series and their (formal) derivatives at the boundary of the convergence circle, obtaining in particular a theorem of Bugajewski and Gan concerning the structure of the set of points where a formal power series is convergent with all its derivatives. ...

A note on formal power series

Xiao-Xiong Gan, Dariusz Bugajewski (2010)

Commentationes Mathematicae Universitatis Carolinae

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In this note we investigate a relationship between the boundary behavior of power series and the composition of formal power series. In particular, we prove that the composition domain of a formal power series g is convex and balanced which implies that the subset 𝕏 ¯ g consisting of formal power series which can be composed by a formal power series g possesses such properties. We also provide a necessary and sufficient condition for the superposition operator T g to map 𝕏 ¯ g into itself or to...

On Kaluza's sign criterion for reciprocal power series

Árpád Baricz, Jetro Vesti, Matti Vuorinen (2011)

Annales UMCS, Mathematica

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T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzyż is applied.

On Kaluza’s sign criterion for reciprocal power series

Arpad Baricz, Jetro Vesti, Matti Vuorinen (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzyż is applied.