A review of definitions of fractional differences and sums

Qiushuang Wang; Run Xu

doi:10.3934/mfc.2022013

Article Contents

2023, Volume 6, Issue 2: 136-160. Doi: 10.3934/mfc.2022013

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A review of definitions of fractional differences and sums

Qiushuang Wang and
Run Xu^,

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

^*Corresponding author: Run Xu
^*Corresponding author: Run Xu

Received: January 2022

Revised: April 2022

Early access: May 2022

Published: May 2023

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Abstract

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.

Keywords:

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

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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 $

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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 $

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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} $

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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]	$\begin{align} & \nabla _{0}^{\alpha }(f(z)g(z))=\sum\limits_{k=0}^{\infty }{\left( \begin{array}{{35}{l}} \alpha \\ k \\ \end{array} \right)}\left( \nabla _{0}^{k}f(z) \right)\nabla _{0}^{\alpha -k}g(z-k) \\ & \Delta _{1}^{\alpha }(f(z)g(z))=\sum\limits_{k=0}^{\infty }{\left( \begin{array}{{35}{l}} \alpha \\ k \\ \end{array} \right)}\left( \Delta _{1}^{k}f(z) \right)\Delta _{1}^{\alpha -k}g(z+k) \\ & \Delta _{\frac{1}{2}}^{\alpha }(f(z)g(z))=\sum\limits_{k=0}^{\infty }{\left( \begin{array}{*{35}{l}} \alpha \\ k \\ \end{array} \right)}\left( \Delta _{\frac{1}{2}}^{k}f(z+\alpha -k) \right)\Delta _{\frac{1}{2}}^{\alpha -k}g(z-k) \\ \end{align}$

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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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