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 .


Introduction
The power series method is a classical tool to approximate solutions to initial value problems for ordinary differential equations.Since fractional calculus became an useful instrument for modeling various phenomena in science and engineering, a lot of classical notions were extended to the fractional case (see [1][2][3][4][5]).For example, the classical power series were generalized to α-fractional power series (with α a positive number) and some classical methods in calculus were extended to the fractional case (see [6]).
The α-fractional power series are used to approximate solutions to fractional ordinary differential equations (FODE).For instance, in [7,8], the solutions to the Bagley-Torvik equation and the fractional Laguerre-type logistic equation are approximated by using α-fractional power series.Numerical approximations of solutions to fractional ordinary differential equations using α-fractional power series can be found in [9][10][11] and references therein.Results on the solutions to systems of fractional ordinary differential equations are presented in [12].A generalization of the α-fractional power series is studied in [13], and is applied to obtain solutions to linear fractional order differential equations.The theoretical background and the applications of the fractional-calculus operators which are based upon the general Fox-Wright function and its special forms as Mittag-Leffler-type functions are presented in [14,15].
The research in the field of fractional differential equations has focused mostly on initial value problems, but there are also some papers dealing with boundary value problems (see [16][17][18][19]).For instance, in [17,18], the existence and uniqueness of a solution to the boundary value problems for fractional order differential equations and nonlocal boundary condition are studied.In [19], the authors use the fractional central formula, based on the generalized Taylor theorem [20], for approximating the fractional derivatives of order α and 2α, respectively.
In this paper, we introduce the α-fractional power moduli series as a generalization of the α-fractional power series.We study the properties of these series in Section 2, using sequential fractional derivatives (Theorems 1 and 2).Using the generalized Taylor's formula, sufficient conditions for a function to be represented as an α-fractional power moduli series are established in Corollary 2.
A practical method to approximate solutions to boundary value problems for FODE using the partial sums of α-fractional power series is presented in Section 3 and is applied in some illustrative examples.The α-fractional analytic functions (on an open interval I) are studied in Section 4. The real analytic functions (obtained for α = 1) seem to be a particular case of α-fractional analytic functions, but it is proved (see [6]) that a function representable as an α-fractional power series at a point x 0 , that is, for all x ∈ [x 0 , x 0 + r), must be a real analytic function on the open interval (x 0 , x 0 + r).As a consequence, non-constant α-analytic functions exist only for α = 1 m with m a positive integer and they are exactly the real analytic functions on the interval I.

Fractional Power Series
A series of the form with a n ∈ R and α ∈ (0, 1] is called an α-fractional power series about x 0 .We note that any series of the form where ⌈x⌉ = min{z ∈ Z : z ≥ x} denotes the ceiling function.
Similarly, a series of the form with a n ∈ R and α ∈ (0, 1] is called an α-fractional power moduli series about x 0 . Fractional power series can be studied using the fractional differential and fractional integral operators.We shortly present the most important definitions and results in fractional calculus (see [1][2][3][4][5]).Moreover, starting from these classical results, we introduce a general frame which is needed in the case of fractional power moduli series.
Definition 2. Let I be a real interval, I = (x 0 , b] and f : I → R be a function of class C ρ,x 0 + (I), with ρ > −1.Then, for any x ∈ I, the left-sided Riemann-Liouville fractional integral of order α > 0 of f is defined as If I = [a, x 0 ) is a real interval and f : I → R is a function of class C ρ,x 0 − (I), with ρ > −1, then the right-sided Riemann-Liouville fractional integral of order α > 0 of f is defined as Lemma 1.Let I = [a, b] be a real interval, x 0 ∈ (a, b), α > 0, and f : I → R be a function of class C ρ,x 0 ± (I), where ρ > −1 and ρ ≥ −α.Then, there exist the following limits, and they are finite and equal: . Thus, the Riemann-Liouville fractional integral of order α can be defined on both sides of x 0 by the following continuous function: , where g(x) is a continuous function, we can write: From the Mean Value Theorem, it follows that there exists ξ x ∈ (x 0 , x) such that and we obtain In a similar way, it can be proved that the limit (J α Introduced by M. Caputo in 1967 (see [21]), the fractional derivative operator expressed by Definition 3 can definitely share some similarities with the fractional derivatives considered by J. Liouville in 1832 (see [22], p. 10, formula (B)).That is why recent studies refer to the Caputo fractional derivative as the Liouville-Caputo fractional derivative (see [14][15][16][17]).We thank the reviewer who brought this issue to our attention.ρ,x 0 + (I) with ρ > −1.For any x ∈ I, the left-sided Liouville-Caputo fractional derivative of order α of f is defined as ρ,x 0 − (I) with ρ > −1, then, for any x ∈ I, the right-sided Liouville-Caputo fractional derivative of order α of f is defined as and they are finite and equal.The Liouville-Caputo derivative of f is defined on both sides of x 0 by the continuous function: A remarkable property of the Riemann-Liouville fractional integral operators is the "semigroup property" ([1], Theorem 2.4): if J α x 0 is any one of the operators J α x 0 + , J α x 0 − , and J α x 0 ± , then, for any α, β > 0 and for any suitable function f , we have It follows that, for any α > 0 and n ∈ N, one can write The equality above does not hold in the case of Liouville-Caputo fractional differential operators.Let us take, for instance, the function x .
Let D α x 0 be one of the Liouville-Caputo fractional differential operators D α x 0 + , D α x 0 − , and D α x 0 ± , and n be a positive integer.We denote by Dnα x 0 f the sequential fractional derivative of order n of the function f : As noted above, Dnα For any positive integer n and α ∈ (0, 1), we denote by Ĉn,α  x 0 ± (I) (resp.Ĉn,α x 0 + (I)) the set of all the functions possessing sequential fractional derivatives of order k, Dkα x 0 ± f (resp.Dkα x 0 + f ), which are continuous on Ī, for every k ≤ n.
that is, the graph of f is symmetric with respect to the straight line x = x 0 .Then, for any ρ > −1 and α ∈ (0, 1) we have: ρ,x 0 − (x 0 − r, x 0 ), and Proof.(i) First of all, we notice that the function f satisfies (3) if and only if there is a continuous function h : (0, r) → R such that Obviously, we have and for any x ∈ (0, r).We can write In a similar way, it can be proved that (D α for all x ∈ (0, r) and so the lemma is proved.
By Lemma 2, it follows that The following theorem establishes the basic properties of the α-fractional power moduli series and extends the results from [6,23] regarding fractional power series.
Proof.(i) Let us consider the power series v(t) = ∞ ∑ n=0 a n t n , where t = |x − x 0 | α .Then, the statement follows by the well-known properties of the power series and the definition of f .(ii) Let h : [0, r) → R be the sum of the fractional power series Then, h is continuous and there exists the fractional Liouville-Caputo derivative D α 0+ h : [0, r) → R (see [23], Theorem 1).Moreover, the series of the fractional derivatives x nα is absolutely and uniformly convergent on [0, b], for any b ∈ (0, r) and x (n−1)α , for all x ∈ [0, r).
We note that the operator D α , defined for power series is known as the Gelfond-Leontiev operator with respect to the Mittag-Leffler function [1,24].Using this operator, the Liouville-Caputo fractional derivative of the function ( 8) can be written as Corollary 1. Assume that the series (2) has a positive radius of convergence r.Then, for every non-negative integer k, there exists the sequential fractional derivative of order k ( Dkα x 0 ± f ), which is a continuous function on I = (x 0 − r, x 0 + r) and Moreover, a n = ( Dnα x 0 ± f )(x 0 ) Γ(nα+1) , for every n ≥ 0.
Proof.The relation (10) follows by applying Theorem 2 k times.By taking x = x 0 in (10), we get the last statement.
The following corollary provides a sufficient condition for a function satisfying (3) to be represented as a fractional power moduli series.
|, then f is represented as an α-fractional power moduli series on I: |x − x 0 | nα , for all x ∈ I.
Proof.By Theorem 1, we get which implies the corollary.

Boundary Value Problems for Fractional Differential Equations
In this section, we present a method to study the existence and the uniqueness of solutions to boundary value problems for fractional linear differential equations, solutions which are representable as α-fractional power series.This is based on the result below.
In order to prove that the boundary value problem (12), ( 13) has a (unique) solution representable as an α-fractional power series at 0 on the interval [0, b], we firstly solve an initial value problem for the same equation (see, for example, [26], p. 88).
Le us assume that y = y(x) is represented as a fractional power series at x 0 = 0 on an interval I = [0, b ′ ), with b ′ > b.Thus, by Theorem 2, we can write for all x ∈ I: If the function y given by ( 14) is a solution to the fractional differential Equation ( 12), then , for all n ≥ 0, where a −1 = 0, c 0 = c 2 = 1, c 1 = −1, and c n = 0, for all n ≥ 3.
Let us consider the initial value problem for Equation (13) with the initial conditions where s is a real parameter.By Theorem 3, it follows that, for any fixed s, the initial value problem (12), ( 17) has a unique solution ỹ = ỹ(x, s) which can be represented as an α-fractional power series at 0 on [0, b ′ ): As shown above, the coefficients ãn (s) must verify (16), and, from the initial conditions (17), we have ã0 (s) = δ 1 and ã1 (s) = s Γ(α+1) .Thus, by (16), we find , for all n ≥ 5.
Hence, for every k ≥ 1, it follows that .
We notice that the coefficients ã3k (s) and ã3k+2 (s) do not depend on s.For every n ≥ 3, we denote otherwise, . By (19), we get , for all n ≥ 4.
By replacing in (18), we obtain It can be easily proved that the fractional power series in the formula (20) have the radius of convergence r = ∞ (hence, they are uniformly convergent on [0, b]).We denote the sum of the series by g To obtain a solution to the boundary value problem (12), (13), we need to find the value of s for which ỹ(b, s) = δ 2 , that is, to solve the equation Hence, for the series (18) is a solution to the boundary value problem (12), (13).
Let us suppose that y 1 (x) is another solution to the boundary value problem (12), ( 13) which can be represented as an α-fractional power series at 0 on [0, b] and denote s 1 = ( Dα 0+ ỹ)(0).Then, the function y 1 (x), is the solution to the initial value problem for Equation (12).Since y 1 (x) satisfies the boundary condition (13) and s 0 is uniquely defined by (21), it follows that s 1 = s 0 .Hence y 1 (x) = ỹ(x, s 0 ), which implies the uniqueness of the solution to the boundary value problem (12), (13).
To obtain a solution to the boundary value problem (22), (23) we have to find s such that .
The uniqueness of the solution follows as in Example 1.

Fractional Analytic Functions
A function is said to be representable as an α-fractional power series at x 0 if it is equal to the sum of the fractional Taylor series about x 0 on an interval [x 0 , x 0 + r).Some authors (see [2], Definition 7.8) call such functions α-analytic at x 0 .In the following, we define the α-analytic functions on an open interval.Definition 4. Let I be an open interval and α ∈ (0, 1].A real function f defined on I is called α-analytic on I, if, for every x 0 ∈ I, there exists ε > 0 such that J x 0 ,ε = [x 0 , x 0 + ε) ⊂ I and f can be represented as an α-fractional power series at x 0 , f (x) = ∞ ∑ n=0 a n (x − x 0 ) nα , for all x ∈ J x 0 ,ε .Remark 2. By Definition 4, it follows that f is an α-fractional analytic on I, if, for every x 0 ∈ I, there exists ε > 0 such that Jx 0 ,ε = (x 0 − ε, x 0 + ε) ⊂ I and the real function S x 0 ,ε ( f ) defined by can be represented as an α-fractional power moduli series (2) at x 0 on Jx 0 ,ε .
If α = 1 in Definition 4, then the classical real analytic functions are obtained.A well-known result (see [27], Corollary 1.2.4) establishes that the sum of a power series, f (x) = ∑ n≥0 a n (x − x 0 ) n , is analytic on the interval (x 0 − r, x 0 + r), where r is the radius of convergence of the series.In the following, we discuss the analyticity of the functions representable as fractional power series.
Obviously, an (integer) power series can be also considered as an α-fractional power series, for any α = 1 m , m ∈ N * .Hence, any real analytic function on an open interval I is also an α-analytic function on I (with α = 1 m ).By Theorem 4, it follows that any α-analytic function on I is real analytic on I and, from Remark 1, we obtain that α must be a rational number if f is non-constant.On the other hand, if α = k m with k > 1, then, for any x 0 ∈ I, we have f (x) = ∞ ∑ n=0 f (kn) (x 0 ) (kn)! (x − x 0 ) kn in a neighbourhood of x 0 , so f ′ (x 0 ) = 0 for all x 0 ∈ I and the next corollary follows.m , m ∈ N * , and f is real analytic on I.

Conclusions
In this paper, we study the properties of the α-fractional power moduli series as a generalization of the α-fractional power series.Using the generalized Taylor's formula with fractional derivatives, a sufficient condition for a function to be represented as an α-fractional power moduli series is established.
Moreover, we present a practical method to solve the boundary value problems for fractional differential equations, the solution being expressed as an α-fractional power series.
Finally, α-analytic functions are defined and we prove that a non-constant function is α-analytic on an open interval I if and only if α = 1 m with m a positive integer, and the function is real analytic on I.

Definition 3 .
Let α > 0 and m = ⌈α⌉.Consider the interval I = (x 0 , b], and f : I → R a function of class C (m) the radius of convergence of the series (2).If r > 0, then (i) For any b ∈ (0, r), the series (2) converges absolutely and uniformly on [x 0 − b, x 0 + b], and there exists a positive integer n(b) such that |a n | ≤ b −nα , for all n ≥ n(b).If f

Figure 1 .
Figure 1.Solutions to BVP in Example 1 for different values of α.

Figure 2 .
Figure 2. Solutions to BVP in Example 2 for different values of α.

Corollary 3 .
Let I be an open interval and f : I → R be a non-constant function.Then, f is an α-analytic function if and only if α = 1