Some Properties of the Functions Representable as Fractional Power Series

The α-fractional power moduli series are introduced as a generalization of α-fractional power series and the structural properties of these series are investigated. Using the fractional Taylor’s formula, sufficient conditions for a function to be represented as an α-fractional power moduli series are established. Beyond theoretical formulations, a practical method to represent solutions to boundary value problems for fractional differential equations as α-fractional power series is discussed. Finally, α-analytic functions on an open interval I are defined, and it is shown that a non-constant function is α-analytic on I if and only if 1/α is a positive integer and the function is real analytic on I.