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Article Contents

# A review of definitions of fractional differences and sums

• *Corresponding author: Run Xu
• Given the increasing importance of discrete fractional calculus in mathematics, science engineering and so on, many different concepts of fractional difference and sum operators have been defined. In this paper, we mainly reviews some definitions of fractional differences and sum operators that emerged in the fields of discrete calculus. Moreover, some properties of those operators are also analyzed and compared with each other, including commutation rules, linearity, Leibniz rules, etc.

Mathematics Subject Classification: Primary: 47E05; Secondary: 34L30.

 Citation:

• Table 1.  Notation List

 Symbol Meaning $\mathbb{C}$ The complex numbers $\mathbb{R}$ The real numbers $\mathbb{R}^{+}$ The positive real numbers $\mathbb{Z}$ The integers: $\mathbb{Z}=\left\lbrace n | n=\cdots,-2,-1,0,1,2,\cdots\right\rbrace$ $\mathbb{Z}^{-}$ The negative integers: $\mathbb{Z}^{-}=\left\lbrace n | n=-1,-2,\cdots\right\rbrace$ $\mathbb{N}$ The natural numbers: $\mathbb{N}=\left\lbrace n | n=0,1,2,\cdots\right\rbrace$ $\mathbb{N}^{+}$ The positive integers: $\mathbb{N}^{+}=\left\lbrace n | n=1,2,\cdots\right\rbrace$ $\mathbb{N}_{a}$ $\mathbb{N}_{a}=\left\lbrace a,a+1,a+2,\cdots | a\in\mathbb{R}\right\rbrace$ $_{b}\mathbb{N}$ $_{b}\mathbb{N}=\left\lbrace b,b-1,b-2,\cdots | b\in\mathbb{R}\right\rbrace$ $\mathbb{N}^{b}_{a}$ $\mathbb{N}^{b}_{a}=\left\lbrace a,a+1,a+2,\cdots ,b| a\in\mathbb{R}, b\in\mathbb{R}, a < b\right\rbrace$ $\mathbb{N}_{a,h}$ $\mathbb{N}_{a,h}=\left\lbrace a,a+h,a+2h,\cdots\right\rbrace$ $_{b,h}\mathbb{N}$ $_{b,h}\mathbb{N}=\left\lbrace b,b-h,b-2h,\cdots\right\rbrace$ $\mathcal{R}$ $\mathcal{R}=\left\lbrace p:\mathbb{N}_{a}\to\mathbb{R}\ {\rm such that}\ 1+p(t)\ne0\ {\rm for}\ t\in\mathbb{N}_{a}\right\rbrace$ $\mathbb{T}$ A time scale $h$ The step of time scale ($h >0$), generally $h=1$

Table 2.  Commutation rules and Linearity for some operators

 Fractional difference/sum Commutation rules Linearity (11)[28] $\circ$[45] $\checkmark$ (12)[22] $\circ$[23] $\checkmark$ (13)[34] $\times$ $\checkmark$ (14)[41] $\times$ $\checkmark$ (15)[36] $\checkmark$[20] $\checkmark$ (17)[47] $\checkmark$ $\checkmark$ (21)[9] $\checkmark$ $\checkmark$ (23)[1] $\checkmark$ [4] $\checkmark$ (25)[1] $\checkmark$ [4] $\checkmark$ (31)[42] $\checkmark$ $\checkmark$ (32)[42] $\checkmark$ $\checkmark$ (44)[26] $\times$ $\checkmark$ (86)[12] $\checkmark$ $\checkmark$ (99)[37] $\checkmark$ $\checkmark$

Table 3.  Taylor's formula

 Fractional difference/sum Taylor's formula (33)[14] \begin{align} & f(t)=\sum\limits_{k=0}^{m-1}{\frac{{{(t-a)}^{(k)}}}{k!}}{{\Delta }^{k}}f(a)+\frac{1}{\Gamma (\alpha )}\sum\limits_{s=a+(m-\alpha )}^{t-\alpha }{{{(t-\sigma (s))}^{(\alpha -1)}}}\ _{a}^{C}{{\Delta }^{\alpha }}f(s), \\ & \alpha \in {{\mathbb{R}}^{+}},\ m=\left\lceil \alpha \right\rceil >0,\ t\in {{\mathbb{N}}_{a+m}},a\in \mathbb{N} \\ \end{align} (35)[15] \begin{align} & f(t)=\sum\limits_{k=0}^{m-1}{\frac{{{(t-a)}^{{\bar{k}}}}}{k!}}{{\nabla }^{k}}f(a)+\frac{1}{\Gamma (\alpha )}\sum\limits_{\tau =a+1}^{t}{{{(t-\tau +1)}^{\overline{\alpha -1}}}}\ _{a}^{C}{{\nabla }^{\alpha }}f(\tau ), \\ & t\in \mathbb{Z},\ t\le a+m,\ m=\left\lceil \alpha \right\rceil >0 \\ \end{align} (36)[17] \begin{align} & f(t)=\sum\limits_{k=0}^{m-1}{\frac{{{(t-b)}^{{\bar{k}}}}}{k!}}{{\nabla }^{k}}f(b)+\frac{1}{\Gamma (\alpha )}\sum\limits_{s=t+\alpha }^{b-v}{{{(s-t-1)}^{(\alpha -1)}}}{{(}^{C}}\nabla _{b}^{\alpha }f)(s), \\ & \alpha \in {{\mathbb{R}}^{+}},\ m=\left\lceil \alpha \right\rceil ,\ v=m-\alpha ,\ t\in {{\ }_{b-m}}\mathbb{N} \\ \end{align} (38)[6] \begin{align} & f(t)=\sum\limits_{k=0}^{m-1}{\frac{{{(b-t)}^{(k)}}}{k!}}{{(-1)}^{k}}{{\nabla }^{k}}f(b)+\frac{1}{\Gamma (\alpha )}{{\sum\limits_{s=t+\alpha }^{b-(m-\alpha )}{{{(\rho (s)-t)}^{(\alpha -1)}}}}^{C}}\nabla _{b}^{\alpha }f(s) \\ & \alpha \in {{\mathbb{R}}^{+}},m=\left\lceil \alpha \right\rceil >0,\quad t{{\in }_{b-m}}\mathbb{N},b\in \mathbb{N} \\ \end{align} (40)[2] $f(t)=\sum\limits_{k=0}^{n-1} \frac{(t-a)^{\overline{k}}}{k!} \nabla^{k}f(a)+\frac{1}{\Gamma(\alpha)}\sum\limits_{k=a}^{t}(t-k+1)^{\overline{\alpha-1}}\ ^{C} _{a}\nabla^{\alpha}f(k),\ t\in\mathbb{N}_{a}$ (41)[2] f(t)=$\sum\limits_{k=0}^{n-1} \frac{(b-t)^{\overline{k}}}{k!}(-1)^{k}\Delta^{k}f(b)+\frac{1}{\Gamma(\alpha)}\sum\limits_{s=t}^{b-1}(s-\rho(t))^{\overline{\alpha-1}}\ ^{C}\nabla_{b}^{\alpha}f(s),\ t\in\ _{b}\mathbb{N}$

Table 4.  Generalized Leibniz rule

 Fractional difference/sum Generalized Leibniz rule (13)[34] {{\Delta }^{\alpha }}f(x)g(x)=\sum\limits_{k=0}^{\infty }{\left( \begin{matrix} \begin{align} & \alpha \\ & k \\ \end{align} \\ \end{matrix} \right)}{{\Delta }^{\alpha -k}}f(x){{\Delta }^{k}}g(x+\alpha -k),\quad \forall \alpha \in \mathbb{C}\setminus \mathbb{Z} (44)[26] \begin{aligned} &\left({ }_{\gamma} \diamond_{a}^{-\alpha,-\beta} f(t) g(t)\right)(t) \\ &=\gamma \sum_{k=0}^{\infty}\left(\begin{array}{c} -\alpha \\ k \end{array}\right)\left[\left(\nabla^{k} g\right)(t)\right] \cdot\left[\left(\Delta_{a}^{-(\alpha+k)} f\right)(t+\alpha+k)\right] \\ &+(1-\gamma) \sum_{k=0}^{\infty}\left(\begin{array}{c} -\beta \\ k \end{array}\right)\left[\left(\nabla^{k} g\right)(t)\right]\left[\left(\Delta^{-(\beta+k)} f\right)(t+\beta+k)\right] \\ &0<\alpha, \beta<1, t=\mathbb{N}_{a} \end{aligned} (15)[36] \begin{align} & {{\nabla }^{\alpha }}f(t)g(t)=\sum\limits_{k=0}^{\alpha }{\left( \begin{array}{*{35}{l}} \alpha \\ k \\ \end{array} \right)}\left[ {{\nabla }^{\alpha -k}}f(t-k) \right]\left[ {{\nabla }^{k}}g(t) \right],\quad \alpha \in \mathbb{N} \\ & _{a}{{\nabla }^{\alpha }}f(t)g(t)=\sum\limits_{k=0}^{t-a}{\left( \begin{array}{*{35}{l}} \alpha \\ k \\ \end{array} \right)}\left[ t-k\nabla _{a}^{\alpha -k}f(t-k) \right]\left[ {{\nabla }^{k}}g(t) \right],\quad \alpha \notin \mathbb{N} \\ \end{align} (31)(32)[42] \begin{align} & \nabla _{0}^{\alpha }(f(z)g(z))=\sum\limits_{k=0}^{\infty }{\left( \begin{array}{*{35}{l}} \alpha \\ k \\ \end{array} \right)}\left( \nabla _{0}^{k}f(z) \right)\nabla _{0}^{\alpha -k}g(z-k) \\ & \Delta _{1}^{\alpha }(f(z)g(z))=\sum\limits_{k=0}^{\infty }{\left( \begin{array}{*{35}{l}} \alpha \\ k \\ \end{array} \right)}\left( \Delta _{1}^{k}f(z) \right)\Delta _{1}^{\alpha -k}g(z+k) \\ & \Delta _{\frac{1}{2}}^{\alpha }(f(z)g(z))=\sum\limits_{k=0}^{\infty }{\left( \begin{array}{*{35}{l}} \alpha \\ k \\ \end{array} \right)}\left( \Delta _{\frac{1}{2}}^{k}f(z+\alpha -k) \right)\Delta _{\frac{1}{2}}^{\alpha -k}g(z-k) \\ \end{align}

Table 5.  Summation by parts formula

 Fractional difference/sum summation by parts formula (23)(25)[1] $\sum\limits_{s=a+1}^{b-1}g(s)\ _{a}\nabla^{-\alpha}f(s)=\sum\limits_{s=a+1}^{b-1}f(s)\nabla_{b}^{-\alpha}g(s),\ \alpha\in\mathbb{R}^{+}$ (24)(26)[1] $\sum\limits_{s=a+1}^{b-1}f(s)_{a}\nabla^{\alpha}g(s)=\sum\limits_{s=a+1}^{b-1}g(s)\nabla_{b}^{\alpha}f(s),\ \alpha\notin\mathbb{Z}$ (18)(22)[9] \begin{align} & \sum\limits_{s=a+(n-\alpha )-1}^{b-n+1}{f}{{(s)}_{a}}{{\Delta }^{\alpha }}g(s)=\sum\limits_{s=a+n-1}^{b-(n-\alpha )+1}{g}(s)\Delta _{b}^{\alpha }f(s), \\ & \alpha >0,n=\left\lceil \alpha \right\rceil ,b={{\mathbb{N}}_{a+n-\alpha }} \\ \end{align} (17)(21)[4] $\sum\limits_{s=a+1}^{b-1}{g}(s)\left( a+1{{\Delta }^{-\alpha }}f \right)(s+\alpha )=\sum\limits_{s=a+1}^{b-1}{f}(s)\Delta _{b-1}^{-\alpha }g(s-\alpha ),\alpha >0,b\in {{\mathbb{N}}_{a}}$ (18)(22)[4] \begin{align} & \sum\limits_{s=a+1}^{b-1}{f}{{(s)}_{a+1}}{{\Delta }^{\alpha }}g(s-\alpha )=\sum\limits_{s=a+1}^{b-1}{g}(s)\Delta _{b-1}^{\alpha }f(s+\alpha ), \\ & \alpha \in {{\mathbb{R}}^{+}},\alpha \notin \mathbb{Z},b\in {{\mathbb{N}}_{a}} \\ \end{align} (40)(41)[2] \begin{align} & \sum\limits_{s=a+1}^{b-1}{g}(s)\mathbb{R}_{a}^{C}{{\nabla }^{\alpha }}f(s)=\left. f(s)\nabla _{b}^{-(1-\alpha )}g(s) \right|_{a}^{b-1}+\sum\limits_{s=a}^{b-2}{f}(s)\nabla _{b}^{\alpha }g(s) \\ & =f(s)g(b-1)-f(s)\nabla _{b}^{-(1-\alpha )}g(a)+\sum\limits_{s=a}^{b-2}{f}(s)\nabla _{b}^{\alpha }g(s) \\ & 0<\alpha <1,a={{\mathbb{N}}_{b}} \\ \end{align} (33)(38)[2] \begin{align} & \sum\limits_{s=a+1}^{b+1}{g}(s)_{a}^{C}{{\Delta }^{\alpha }}f(s-\alpha )=\left. f(s)\Delta _{b-1}^{-(1-\alpha )}g(s-(1-\alpha )) \right|_{a}^{b-1} \\ & +\sum\limits_{s=a}^{b-2}{f}(s)\Delta _{b-1}^{\alpha }g(s+\alpha ),0<\alpha <1,a={{\mathbb{N}}_{b}} \\ \end{align} (17)(21)[9] $\sum\limits_{s=a+\alpha }^{b}{\left( _{a}{{\Delta }^{-\alpha }}f \right)}(s)g(s)=\sum\limits_{s=a}^{b-\alpha }{f}(s)\Delta _{b}^{-\alpha }g(s),\alpha >0,b={{\mathbb{N}}_{a+\alpha }}$
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