Some Properties of the Functions Representable as Fractional Power Series

The alpha-fractional power moduli series are introduced as a generalization of alpha-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 alpha-fractional power moduli series are established. Beyond theoretical formulations, a practical method to represent solutions to boundary value problems for fractional differential equations as alpha-fractional power series is discussed. Finally, alpha-analytic functions on an open interval I are defined, and it is shown that a non-constant function is alpha-analytic on I if and only if 1/alpha is a positive integer and the function is real analytic on I.