Qualitative Theory of Dynamical Systems

, Volume 16, Issue 1, pp 71–100 | Cite as

Generalizing Taylor Expansion Series Through Succeeding Initial Value Problems



In this paper, the classical Taylor’s expansion series for a given continuous and k-times differentiable real function is obtained as the unique solution of a certain class of initial value problems. Further, through some subsequent generalizations regarding that problem in terms of certain derivative-based operators, we obtain some generalized Taylor’s type polynomial expansions, including the Taylor–Aleph series, which remains as particular cases. In addition to that, some analytical properties about these involved operators are also provided.


Initial value problem Taylor expansion series Laplace transform 

Mathematics Subject Classification

Primary 41A58 Secondary 26E10 26A06 30K05 



The authors would like to thank J.J. Rodríguez Cano for several fruitful discussions and useful hints.


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Copyright information

© Springer International Publishing 2015

Authors and Affiliations

  1. 1.University Centre of Defence at the Spanish Air Force Academy, MDE-UPCTMurciaSpain
  2. 2.Departamento de Matemática Aplicada y Estadística, Hospital de MarinaUniversidad Politécnica de CartagenaCartagenaSpain

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