Abstract
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.