# New Results on Fractional Power Series: Theories and Applications

^{1}

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^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notations on Fractional Calculus Theory

**Definition 2.1:**A real function f(x), x > 0 is said to be in the space C

_{μ}, μ ∈ ℝ if there exists a real number ρ > μ such that f(x) = x

^{ρ}f

_{1}(x), where f

_{1}(x) ∈ C[0, ∞), and it is said to be in the space ${C}_{\mu}^{n}$ if f

^{(n)}(x) ∈ C

_{μ}, n ∈ ℕ.

**Definition 2.2:**The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function f(x) ∈ C

_{μ}, μ ≥ −1 is defined as:

_{μ}, μ ≥ −1, α, β ≥ 0, C ∈ ℝ, and γ ≥ −1, we have ${J}_{s}^{\alpha}{J}_{s}^{\beta}f\left(x\right)={J}_{s}^{\alpha +\beta}f\left(x\right)={J}_{s}^{\beta}{J}_{s}^{\alpha}f\left(x\right)$, ${J}_{s}^{\alpha}C=\frac{C}{\Gamma \left(\alpha +1\right)}{\left(x-s\right)}^{\alpha}$, and ${J}_{s}^{\alpha}{\left(x-s\right)}^{\gamma}=\frac{\Gamma \left(\gamma +1\right)}{\Gamma \left(\alpha +\gamma +1\right)}{\left(x-s\right)}^{\alpha +\gamma}$.

**Definition 2.3:**The Riemann-Liouville fractional derivative of order α > 0 of $f\in {C}_{-1}^{n},n\in \mathbb{N}$ is defined as:

**Definition 2.4:**The Caputo fractional derivative of order α > 0 of $f\in {C}_{-1}^{n},n\in \mathbb{N}$ is defined as:

**Lemma 2.1:**If $n-1<\alpha \le n$, $f\in {C}_{\mu}^{n}$, $n\in \mathbb{N}$, and $\mu \ge -1$, then ${D}_{s}^{\alpha}{J}_{s}^{\alpha}f\left(x\right)=f\left(x\right)$ and ${J}_{s}^{\alpha}{D}_{s}^{\alpha}f\left(x\right)=f\left(x\right)-{\displaystyle \sum}_{j=0}^{n-1}{f}^{\left(j\right)}\left({s}^{+}\right)\frac{{\left(x-s\right)}^{j}}{j!}$, where $x>s\ge 0$.

## 3. Fractional Power Series Representation

**Definition 3.1:**A power series representation of the form

_{0}is called a FPS about t

_{0}, where t is a variable and C

_{n}’s are constants called the coefficients of the series.

_{0}= 0 the expansion $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha$ is called a fractional Maclaurin series. Notice that in writing out the term corresponding to n = 0 in Equation (8) we have adopted the convention that (t − t

_{0})

^{0}= 1 even when t = t

_{0}. Also, when t = t

_{0}each of the terms of Equation (8) vanishes for n ≥ 1 and so. On the other hand, the FPS (8) always converges when t = t

_{0}. For the sake of simplicity of our notation, we shall treat only the case where t

_{0}= 0 in the first four theorems. This is not a loss of the generality, since the translation t’ = t − t

_{0}reduces the FPS about t

_{0}to the FPS about 0.

**Theorem 3.1:**We have the following two cases for the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha},t\ge 0$:

- (1)
- If the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha$ converges when $t=b>0$, then it converges whenever $0\le t<b$,
- (2)
- If the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha$ diverges when $t=d>0$, then it diverges whenever $t>d$.

**Proof:**For the first part, suppose that ${{\displaystyle \sum}}^{\text{}}{c}_{n}{b}^{n\alpha}$ converges. Then, we have $\underset{n\to \infty}{\mathrm{lim}}{c}_{n}{b}^{n\alpha}=0$. According to the definition of limit of sequences with $\epsilon =1$, there is a positive integer $N$ such that $\left|{c}_{n}{b}^{n\alpha}\right|<1$ whenever $n\ge N$. Thus, for $n\ge N$, we have $\left|{c}_{n}{t}^{n\alpha}\right|=\left|\frac{{c}_{n}{b}^{n\alpha}{t}^{n\alpha}}{{b}^{n\alpha}}\right|=\left|{c}_{n}{b}^{n\alpha}\right|{\left|\frac{t}{b}\right|}^{n\alpha}<{\left|\frac{t}{b}\right|}^{n\alpha}$. Again, if $0\le t<b$ , then ${\left|\frac{t}{b}\right|}^{\alpha}<1$, so ${{\displaystyle \sum}}^{\text{}}{\left|\frac{t}{b}\right|}^{n\alpha}$ is a convergent geometric series. Therefore, by the comparison test, the series $\sum}_{n=N}^{\infty}\left|{c}_{n}{t}^{n\alpha}\right|$ is convergent. Thus the series ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ is absolutely convergent and therefore convergent. To prove the remaining part, suppose that ${{\displaystyle \sum}}^{\text{}}{c}_{n}{d}^{n\alpha}$ diverges. Now, if $t$ is any number such that $t>d>0$, then ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ cannot converge because, by Case 1, the convergence of ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ would imply the convergence of ${{\displaystyle \sum}}^{\text{}}{c}_{n}{d}^{n\alpha}$. Therefore, ${{\displaystyle \sum}}^{\text{}}{c}_{n}{d}^{n\alpha}$ diverges whenever $t>d$. This completes the proof.

**Theorem 3.2:**For the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha},t\ge 0$, there are only three possibilities:

- (1)
- The series converges only when $t=0$,
- (2)
- The series converges for each $t\ge 0$,
- (3)
- There is a positive real number $R$ such that the series converges whenever $0\le t<R$ and diverges whenever $t>R$.

**Proof:**Suppose that neither Case 1 nor Case 2 is true. Then, there are nonzero numbers $b$ and $d$ such that ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ converges for $t=b$ and diverges for $t=d.$ Therefore, the set $S=\{t|{{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ converges } is not empty. By the preceding theorem, the series diverges if $t>d$, so $0\le t\le d$ for each $t\in S$. This says that $d$ is an upper bound for $S$. Thus, by the completeness axiom, $S$ has a least upper bound $R$. If $t>R$, then $t\notin S$, so ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ diverges. If $0\le t<R$, then $t$ is not an upper bound for $S$ and so there exists $b\in S$ such that $b>t$. Since $b\in S$ and ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ converges, so by the preceding theorem ${{\displaystyle \sum}}^{\text{}}{c}_{n}{t}^{n\alpha}$ converges, so the proof of the theorem is complete.

**Remark 3.1:**The number $R$ in Case 3 of Theorem 3.2 is called the radius of convergence of the FPS. By convention, the radius of convergence is$R=0$ in Case 1 and $R=\infty $ in Case 2.

**Theorem 3.3:**The CPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n},-\infty t\infty $ has radius of convergence$R$ if and only if the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha},t\ge 0$ has radius of convergence${R}^{1/\alpha}$.

**Proof:**If we make the change of variable t = x

^{α}, x ≥ 0 then the CPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n$ becomes $\sum}_{n=0}^{\infty}{c}_{n}{x}^{n\alpha$. This series converges for 0 ≤ x

^{α}< R, that is for 0 ≤ x < R

^{1/α}, and so the FPS $\sum}_{n=0}^{\infty}{c}_{n}{x}^{n\alpha$ has radius of convergence R

^{1/α}. Conversely, if we make the change of variable t = x

^{1/α}, x ≥ 0 then the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha$ becomes $\sum}_{n=0}^{\infty}{c}_{n}{x}^{n$. In fact, this series converges for 0 ≤ x

^{1/α}< R

^{1/α}that is for 0 ≤ x < R. Since the two series $\sum}_{n=0}^{\infty}{c}_{n}{x}^{n},x\ge 0$ and $\sum}_{n=0}^{\infty}{c}_{n}{x}^{n},$ −∞ < x < ∞ have the same radius of convergence $R=\underset{n\to \infty}{\mathrm{lim}}\left|\frac{{c}_{n}}{{c}_{n+1}}\right|$, the radius of convergence for the CPS $\sum}_{n=0}^{\infty}{c}_{n}{x}^{n},$ −∞ < x < ∞ is R, so the proof of the theorem is complete.

**Theorem 3.4:**Suppose that the FPS $\sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha$ has radius of convergence$R0$. If $f\left(t\right)$ is a function defined by $f\left(t\right)={\displaystyle \sum}_{n=0}^{\infty}{c}_{n}{t}^{n\alpha}$ on 0 ≤ t < R, then for 0 ≤ m − 1 < α < m and 0 ≤ t < R, we have:

**Proof:**Define $g\left(x\right)={\displaystyle \sum}_{n=0}^{\infty}{c}_{n}{x}^{n}$ for 0 ≤ x < R

^{α}, where R

^{α}is the radius of convergence. Then:

^{α}. On the other hand, if we make the change of variable x = t

^{α}, t ≥ 0 into Equation (11) and use the properties of the operator ${D}_{0}^{\alpha}$, we obtain:

^{α}. Similarly, if we make the change of variable x = t

^{α}, t ≥ 0 into Equation (13), we can conclude that:

**Theorem 3.5:**Suppose that f has a FPS representation at t

_{0}of the form:

_{0}, t

_{0}+ R) and$\text{}{D}_{{t}_{0}}^{n\alpha}f\left(t\right)\in C\left({t}_{0},{t}_{0}+R\right)$ for n = 0,1,2,…, then the coefficients c

_{n}in Equation (15) will take the form ${c}_{n}=\frac{{D}_{{t}_{0}}^{n\alpha}f\left({t}_{0}\right)}{\Gamma \left(n\alpha +1\right)}$, where ${D}_{{t}_{0}}^{n\alpha}={D}_{{t}_{0}}^{\alpha}\xb7{D}_{{t}_{0}}^{\alpha}\xb7\dots \xb7{D}_{{t}_{0}}^{\alpha}$ (n-times).

**Proof:**Assume that f is an arbitrary function that can be represented by a FPS expansion. First of all, notice that if we put t = t

_{0}into Equation (15), then each term after the first vanishes and thus we get c

_{0}= f(t

_{0}). On the other aspect as well, by using Equation (9), we have:

_{0}≤ t < t

_{0}+ R. The substitution of t = t

_{0}into Equation (16) leads to ${c}_{1}=\frac{{D}_{{t}_{0}}^{\alpha}f\left({t}_{0}\right)}{\Gamma \left(\alpha +1\right)}$. Again, by applying Equation (9) on the series representation in Equation (16), one can obtain that:

_{0}≤ t < t

_{0}+ R. Here, if we put t = t

_{0}into Equation (17), then the obtained result will be ${c}_{2}=\frac{{D}_{{t}_{0}}^{2\alpha}f\left({t}_{0}\right)}{\Gamma \left(2\alpha +1\right)}$. By now we can see the pattern and discover the general formula for c

_{n}. However, if we continue to operate ${D}_{{t}_{0}}^{\alpha}$(∙) n-times and substitute t = t

_{0}, we can get ${c}_{n}=\frac{{D}_{{t}_{0}}^{n\alpha}f\left({t}_{0}\right)}{\Gamma \left(n\alpha +1\right)},n=0,1,2,\dots $. This completes the proof.

_{0}:

**Theorem 3.6:**Suppose that f has a Generalized Taylor's series representation at t

_{0}of the form:

**Proof:**If we make the change of variable $t={\left(x-{t}_{0}\right)}^{1/\alpha}+{t}_{0}$, ${t}_{0}\le x<{t}_{0}+{R}^{\alpha}$ into Equation (19), then we obtain:

_{0}takes the form:

_{n}(t) = f(t) − T

_{n}(t), then R

_{n}(t) is the remainder of the Generalized Taylor's series.

**Theorem 3.7:**Suppose that f(t) ∈ C[t

_{0}, t

_{0}+ R) and ${D}_{{t}_{0}}^{j\alpha}f\left(t\right)\in C\left({t}_{0},{t}_{0}+R\right)$ for j = 0,1,2,…, n+ 1, where 0 < α ≤ 1. Then f could be represented by:

**Proof:**From the certain properties of the operator ${J}_{\alpha}^{\mathrm{\alpha}}$ and Lemma 2.1, one can find that:

**Theorem 3.8:**If $\left|{D}_{{t}_{0}}^{\left(n+1\right)\alpha}f\left(t\right)\right|\le M$ on t

_{0}≤ t ≤ d, where 0 < α ≤ 1, then the reminder R

_{n}(t) of the Generalized Taylor's series will satisfies the inequality:

**Proof:**First of all, assume that ${D}_{{t}_{0}}^{j\alpha}f\left(t\right)$ exist for $j=0,1,2,\dots ,n+1$ and that:

**Theorem 3.9:**Suppose that $f$ has a FPS representation at ${t}_{0}$ of the form

_{0}, t

_{0}+ R)

**Proof:**Let $g\left(t\right)={\displaystyle \sum}_{n=0}^{\infty}{c}_{n}{t}^{n},\text{}\left|t\right|{R}^{\alpha}$ and h(t) = (t − t

_{0})

^{α}, t

_{0}≤ t < t

_{0}+ R, 0 ≤ m − 1 < α ≤ m. Then g(t) and h(t) are analytic functions and thus the composition (g ∘ h)(t) = f(t) is analytic in (t

_{0}, t

_{0}+ R). This completes the proof.

## 4. Application I: Approximation Fractional Derivatives and Integrals of Functions

**Application 4.1:**Consider the following non-elementary function:

^{(n)}(0) = n!. In other words, the fractional Maclaurin series of $f\left(t\right)$ can be written as $\sum}_{n=0}^{\infty}{t}^{n\alpha},\text{}\alpha 0,t\ge 0$. In fact, this is a convergent geometric series with ratio ${t}^{\alpha}$. Thus, the series is convergent for each $0\le {t}^{\alpha}<1$ and then for each $0\le t<1$. Therefore $f\left(t\right),\text{}0\le t1$ is the sum of its fractional Maclaurin series representation. Note that, this result can be used to approximate the functions ${D}_{0}^{\alpha}f\left(t\right)$ and ${J}_{0}^{\alpha}f\left(t\right)$ on $0\le t<1$ However, according to Equation (9), the function ${D}_{0}^{\alpha}f\left(t\right)$ can be approximated by the $k\text{th}$-partial sum of its expansion as follows:

**Table 1.**The approximate values of ${D}_{0}^{\alpha}f\left(t\right)$ when k = 10 for Application 4.1.

t | α = 0.5 | α = 0.75 | α = 1.5 | α = 2 | ||||
---|---|---|---|---|---|---|---|---|

0 | 0.886227 | 0.919063 | 1.329340 | 2 | ||||

0.1 | 1.448770 | 1.253617 | 1.481250 | 2.123057 | ||||

0.2 | 1.918073 | 1.619507 | 1.814089 | 2.531829 | ||||

0.3 | 2.499525 | 2.113559 | 2.385670 | 3.370618 | ||||

0.4 | 3.261329 | 2.825448 | 3.371164 | 4.994055 | ||||

0.5 | 4.277607 | 3.899429 | 5.179401 | 8.295670 | ||||

0.6 | 5.635511 | 5.569142 | 8.843582 | 15.839711 | ||||

0.7 | 9.803122 | 8.201646 | 17.203979 | 36.370913 | ||||

0.8 | 9.803122 | 12.353432 | 38.389328 | 104.441813 | ||||

0.9 | 12.869091 | 18.839971 | 95.486744 | 365.976156 |

**Table 2.**The approximate values of ${J}_{0}^{\alpha}f\left(t\right)$ when k = 10 for Application 4.1.

t | α = 0.5 | α = 0.75 | α = 1.5 | α = 2 | ||||
---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | ||||

0.1 | 0.121746 | 0.025296 | 0.025296 | 0.000008 | ||||

0.2 | 0.289460 | 0.080361 | 0.001859 | 0.000136 | ||||

0.3 | 0.509120 | 0.165975 | 0.006551 | 0.000700 | ||||

0.4 | 0.795368 | 0.289398 | 0.016399 | 0.002283 | ||||

0.5 | 1.169853 | 0.463570 | 0.034285 | 0.005812 | ||||

0.6 | 1.662260 | 0.709964 | 0.064487 | 0.012745 | ||||

0.7 | 2.311743 | 1.063626 | 0.114028 | 0.025437 | ||||

0.8 | 3.168540 | 1.580987 | 0.195965 | 0.048045 | ||||

0.9 | 4.295695 | 2.351284 | 0.338186 | 0.089143 |

**Application 4.2:**Consider the following Mittag-Leffler function:

_{α}(t

^{α}). Note that ${D}_{0}^{n\alpha}$(E

_{α}(t

^{α})) ∈ C(0, ∞) for n ∈ ℕ and α > 0. In [46] the authors have approximated the function E

_{α}(t

^{α}) for different values of t when 0 < α ≤ 1 by 10-th partial sum of its expansion. However, using Equations (9) and (10) both functions ${D}_{0}^{\alpha}$(E

_{α}(t

^{α})) and ${J}_{0}^{\alpha}$(E

_{α}(t

^{α})) can be approximated, respectively, by the following k-partial sums:

_{α}(t

^{α})) and J(E

_{α}(t

^{α})) for different values of t and α on 0 ≤ t ≤ 4 in step of 0.4 when k = 10.

**Table 3.**The approximate values of ${D}_{0}^{\alpha}$(E

_{α}(t

^{α})) when k = 10 for Application 4.2.

t | α = 0.5 | α = 0.75 | α = 1.5 | α = 2 | ||||
---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | ||||

0.4 | 2.430013 | 1.800456 | 1.201288 | 1.081072 | ||||

0.8 | 3.991267 | 2.816662 | 1.630979 | 1.337435 | ||||

1.2 | 6.220864 | 4.298057 | 2.324700 | 1.810656 | ||||

1.6 | 9.451036 | 6.489464 | 3.389416 | 2.577464 | ||||

2.0 | 14.097234 | 9.743204 | 4.996647 | 3.762196 | ||||

2.4 | 20.683136 | 14.543009 | 7.407121 | 5.556947 | ||||

2.8 | 29.857007 | 21.721976 | 11.012177 | 8.252728 | ||||

3.2 | 42.406132 | 32.250660 | 16.396938 | 12.286646 | ||||

3.6 | 59.270535 | 47.649543 | 24.435073 | 18.312779 | ||||

4.0 | 81.556340 | 69.980001 | 36.430382 | 27.308232 |

t | α = 0.5 | α = 0.75 | α = 1.5 | α = 2 | ||||
---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | ||||

0.4 | 1.430036 | 0.800456 | 0.201288 | 0.081072 | ||||

0.8 | 2.992286 | 1.816664 | 0.630979 | 0.337435 | ||||

1.2 | 5.230333 | 3.298122 | 1.324701 | 0.810656 | ||||

1.6 | 8.497108 | 5.490163 | 2.389416 | 1.577464 | ||||

2.0 | 13.254431 | 8.747610 | 3.996647 | 2.762196 | ||||

2.4 | 20.111627 | 13.592834 | 6.407121 | 4.556947 | ||||

2.8 | 29.857352 | 20.792695 | 10.012177 | 7.252728 | ||||

3.2 | 43.491129 | 31.463460 | 15.396941 | 11.286646 | ||||

3.6 | 62.255682 | 47.211856 | 23.435091 | 17.312779 | ||||

4.0 | 87.670285 | 70.321152 | 35.430382 | 26.308232 |

## 5. Application II: Series Solutions of Fractional Differential Equations

**Application 5.1:**Consider the following linear fractional equation [48]:

_{0}, and ρ

_{0}are real finite constants.

^{nα}to zero in both sides of Equation (37) leads to the following: . Considering the initial conditions (34) one can obtain c

_{0}= y

_{0}and . In fact, based on these results the remaining coefficients of t

^{nα}can be divided into two categories. The even index terms and the odd index terms, where the even index terms take the form and so on, and the odd index term which are , and so on. Therefore, we can obtain the following series expansion solution:

**Application 5.2:**Consider the following composite linear fractional equation [39]:

_{1}and c

_{3}must be zeros. On the other aspect as well, the substituting of the initial conditions (41) into Equation (42) and into ${D}_{0}^{1}$y(t) in Equation (43) gives c

_{0}= 0 and c

_{2}= 0. Therefore, the discretized form of the functions y(t), ${D}_{0}^{\text{1/2}}$y(t), and ${D}_{0}^{2}$y(t) is obtained. The resulting new form will be as follows:

^{n/2}to zero in the resulting equation, and finally identifying the coefficients, we then will obtain recursively the following results: , and . So, the 15th-truncated series approximation of y(t) is:

**Table 5.**The 15th-approximate values of y(t), ${D}_{0}^{\text{1/2}}$y(t), and ${D}_{0}^{2}$y(t) and Res(t) for Application 5.2.

t | y(t) | ${D}_{0}^{\text{1/2}}$y(t) | ${D}_{0}^{2}$y(t) | Res(t) | ||||
---|---|---|---|---|---|---|---|---|

0.0 | 0 | 0 | 0 | 0 | ||||

0.2 | 0.157037 | 0.525296 | 7.317668 | 6.211481 × 10^{−7} | ||||

0.4 | 0.604695 | 1.413213 | 5.982030 | 6.167617 × 10^{−5} | ||||

0.6 | 1.290452 | 2.420120 | 4.288506 | 9.217035 × 10^{−4} | ||||

0.8 | 2.1472288 | 3.409426 | 2.437018 | 6.327666 × 10^{−3} | ||||

1.0 | 3.101501 | 4.282177 | 0.587987 | 2.833472 × 10^{−2} |

**Application 5.3:**Consider the following nonlinear fractional equation [40]:

_{0}must be equal to zero. Therefore:

_{k}= 1 if k = 0 and X

_{k}= 0 if k ≥ 1. From Theorems 3.2 and 3.4, the αkth-derivative of the FPS representation, Equation (48), is convergent at least at t = 0, for k = 0,1,2,…. Therefore, the substituting t = 0 into Equation (51) gives the following recurrence relation which determine the values of the coefficients c

_{n}of t

^{nα}: c

_{0}= 0, ${c}_{1}=\frac{1}{r(\alpha +1)}$, and for k = ,1,2,…. If we collect and substitute these value of the coefficients back into Equation (48), then the exact solution of Equations (46) and (47) has the general form which is coinciding with the general expansion:

**Application 5.4:**Consider the following composite nonlinear fractional equation [40]:

_{0}and c

_{1}are real finite constants.

t | y(t;α = 1.5) | Res(t;α = 1.5) | y(t;α = 2.5) | Res(t;α = 2.5) | ||||
---|---|---|---|---|---|---|---|---|

0.0 | 0 | 0 | 0 | 0 | ||||

0.2 | 0.067330 | 2.034437 × 10^{−17} | 0.005383 | 3.103055 × 10^{−16} | ||||

0.4 | 0.191362 | 4.370361 × 10^{−17} | 0.030450 | 1.252591 × 10^{−15} | ||||

0.6 | 0.356238 | 2.850815 × 10^{−13} | 0.083925 | 7.275543 × 10^{−16} | ||||

0.8 | 0.563007 | 2.897717 × 10^{−10} | 0.172391 | 1.022700 × 10^{−15} | ||||

1.0 | 0.822511 | 6.341391 × 10^{−8} | 0.301676 | 1.998026 × 10^{−16} |

_{0}and c

_{1}are arbitrary, ${c}_{2}=\frac{\Gamma \left(2\alpha +1\right)}{1+{c}_{1}^{2}{\left(\Gamma \left(\alpha +1\right)\right)}^{2}}$ and ${c}_{k+2}=\frac{\Gamma \left(k\alpha +1\right)}{\Gamma \left(\left(2+k\right)\alpha +1\right)}{\displaystyle \sum}_{j=0}^{k}{c}_{j+1}{c}_{k-j+1}\frac{\Gamma \left(\left(j+1\right)\alpha +1\right)}{\Gamma \left(j\alpha +1\right)}\frac{\Gamma \left(\left(k-j+1\right)\alpha +1\right)}{\Gamma \left(\left(k-j\right)\alpha +1\right)}$ for $k=1,2,\dots $. Therefore, by easy calculations we can obtain that the general solution of Equations (54) and (55) agree well with the following expansion:

_{0}and c

_{1}in the set of real or complex numbers.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

El-Ajou, A.; Arqub, O.A.; Zhour, Z.A.; Momani, S.
New Results on Fractional Power Series: Theories and Applications. *Entropy* **2013**, *15*, 5305-5323.
https://doi.org/10.3390/e15125305

**AMA Style**

El-Ajou A, Arqub OA, Zhour ZA, Momani S.
New Results on Fractional Power Series: Theories and Applications. *Entropy*. 2013; 15(12):5305-5323.
https://doi.org/10.3390/e15125305

**Chicago/Turabian Style**

El-Ajou, Ahmad, Omar Abu Arqub, Zeyad Al Zhour, and Shaher Momani.
2013. "New Results on Fractional Power Series: Theories and Applications" *Entropy* 15, no. 12: 5305-5323.
https://doi.org/10.3390/e15125305