doi: 10.3934/mfc.2022013
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A review of definitions of fractional differences and sums

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

*Corresponding author: Run Xu

Received  January 2022 Revised  April 2022 Early access April 2022

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.

Citation: Qiushuang Wang, Run Xu. A review of definitions of fractional differences and sums. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022013
References:
[1]

T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal, 2012 (2012), 13pp. doi: 10.1155/2012/406757.

[2]

T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), 12pp. doi: 10.1155/2013/406910.

[3]

T. Abdeljawad, Different type kernel $h$-fractional differences and their fractional $h$-sums, Chaos Solitons Fractals, 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.

[4]

T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Difference Equ., 2013 (2013), 16pp. doi: 10.1186/1687-1847-2013-36.

[5]

T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Solitons Fractals, 126 (2019), 315-324.  doi: 10.1016/j.chaos.2019.06.012.

[6]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.

[7]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Difference Equ., 2016 (2016), 18pp. doi: 10.1186/s13662-016-0949-5.

[8]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[9]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J Comput. Anal. Appl., 13 (2011), 574-582. 

[10]

T. Abdeljawad, D. Baleanu, F. Jarad and R. P. Agarwal, Fractional sums and differences with binomial coefficients, Discrete Dyn. Nat. Soc., 2013 (2013), 6pp. doi: 10.1155/2013/104173.

[11]

T. Abdeljawad, S. Banerjee and G. -C. Wu, Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption, Optik, 218 (2020). doi: 10.1016/j. ijleo. 2019.163698.

[12]

T. AbdeljawadF. Jarad and J. Alzabut, Fractional proportional differences with memory, Eur. Phys. J. Special Topics, 226 (2017), 3333-3354.  doi: 10.1140/epjst/e2018-00053-5.

[13]

J. O. Alzabut and T. Abdeljawad, Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl., 5 (2014), 177-187. 

[14]

G. A. Anastassiou, Discrete fractional caculus and inequalities, preprint, 2009, arXiv: 0911.3370.

[15]

G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling, 51 (2010), 562-571.  doi: 10.1016/j.mcm.2009.11.006.

[16]

G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Modelling, 52 (2010), 556-566.  doi: 10.1016/j.mcm.2010.03.055.

[17]

G. A. Anastassiou, Right nabla discrete fractional calculus, Int. J. Difference Equ., 6 (2011), 91-104. 

[18]

D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137. 

[19]

F. M. Atıcı and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176. 

[20]

F. M. Atıcı and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., (2009), 1–12. doi: 10.14232/ejqtde. 2009.4.3.

[21]

Z. Bai and R. Xu, The asymptotic behavior of solutions for a class of nonlinear fractional difference equations with damping term, Discrete Dyn. Nat. Soc., 2018 (2018), 11pp. doi: 10.1155/2018/5232147.

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P. Baliarsingh, On a fractional difference operator, Alex. Eng. J., 55 (2016), 1811-1816.  doi: 10.1016/j.aej.2016.03.037.

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P. Baliarsingh and L. Nayak, A note on fractional difference operators, Alex. Eng. J., 57 (2018), 1051-1054.  doi: 10.1016/j.aej.2017.02.022.

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N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Discrete-time fractional variational problems, Signal Process., 91 (2011), 513-524.  doi: 10.1016/j.sigpro.2010.05.001.

[25]

N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., 29 (2011), 417-437.  doi: 10.3934/dcds.2011.29.417.

[26]

N. R. O. Bastos and D. F. M. Torres, Combined delta-nabla sum operator in discrete fractional calculus, Commun. Frac. Calc., 1 (2010), 41-47. 

[27]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.

[28]

S. Chapman, On non-integral orders of summability of series and integrals, Proc. London Math. Soc. (2), 9 (1911), 369-409.  doi: 10.1112/plms/s2-9.1.369.

[29]

F. Chen, X. Luo and Y. Zhou, Existence results for nonlinear fractional difference equation, Adv. Difference Equ., 2011 (2011), 12pp. doi: 10.1155/2011/713201.

[30]

G. V. S. R. Deekshitulu and J. J. Mohan, Fractional difference inequalities, Commun. Appl. Anal., 14 (2010), 89-97. 

[31]

G. V. S. R. Deekshitulu and J. J. Mohan, Some new fractional difference inequalities, in Mathematical Modelling and Scientific Computation, Commun. Comput. Inf. Sci., 283, Springer, Heidelberg, 2012, 403–412. doi: 10.1007/978-3-642-28926-2_44.

[32]

G. V. S. R. Deekshitulu and J. J. Mohan, Some new fractional difference inequalities of Gronwall-Bellman type, Math. Sci., 6 (2012), 9pp.

[33]

E. C. de Oliveira and J. A. Tenreiro Machado, A review of definitions for fractional derivatives and integrals, Math. Probl. Eng., 2014 (2014), 6pp. doi: 10.1155/2014/238459.

[34]

J. B. Diaz and T. J. Osler, Differences of fractional order, Math. Comp., 28 (1974), 185-202.  doi: 10.1090/S0025-5718-1974-0346352-5.

[35]

R. K. GhazianiW. Govaerts and C. Sonck, Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response, Nonlinear. Anal. Real World Appl., 13 (2012), 1451-1465.  doi: 10.1016/j.nonrwa.2011.11.009.

[36]

H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp., 50 (1988), 513-529.  doi: 10.1090/S0025-5718-1988-0929549-2.

[37]

S. S. Haider and M. ur Rehman, On substantial fractional difference operator, Adv. Difference Equ., 2020 (2020), 18pp. doi: 10.1186/s13662-020-02594-0.

[38]

S. S. Haider, M. ur Rehman and T. Abdejawad, On Hilfer fractional difference operator, Adv. Difference Equ., 2020 (2020), 20pp. doi: 10.1186/s13662-020-02576-2.

[39]

R. Hirota, Lectures on difference equations, Science-sha, Japanese, 2000.

[40]

M. Holm, The Theory of Discrete Fractional Calculus Development and Application, Ph. D thesis, University of Nebraska in Lincoln, 2011.

[41]

J. R. M. Hosking, Fractional differencing, Biometrika, 68 (1981), 165-176.  doi: 10.1093/biomet/68.1.165.

[42]

S. C. Jun, A note on fractional differences based on a linear combination between forward and backward differences, Comput. Math. Appl., 41 (2001), 373-378.  doi: 10.1016/S0898-1221(00)00280-7.

[43]

T. Kaczorek, Positive 1D and 2D Systems, Communications and Control Engineering, Springer, London, 2002. doi: 10.1007/978-1-4471-0221-2.

[44]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.

[45]

B. Kuttner, On differences of fractional order, Proc. London Math. Soc. (3), 7 (1957), 453-466.  doi: 10.1112/plms/s3-7.1.453.

[46]

W. N. Li, Oscillation results for certain forced fractional difference equations with damping term, Adv. Difference Equ., 2016 (2016), 9pp. doi: 10.1186/s13662-016-0798-2.

[47]

K. S. Miller and B. Ross, Fractional difference calculus, in Univalent Functions, Fractional Calculus, and Their Applications (Kōriyama, 1988), Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989, 139–152.

[48]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.

[49]

P. O. Mohammeda, T. Abdeljawadb and F. K. Hamasalh, Discrete Prabhakar fractional difference and sum operators, Chaos Solitons Fractals, 150 (2021), 11pp. doi: 10.1016/j. chaos. 2021.111182.

[50]

A. Mouaouine, A. Boukhouima, K. Hattaf and N. Yousfi, A fractional order SIR epidemic model with nonlinear incidence rate, Adv. Difference Equ., 2018 (2018), 9pp. doi: 10.1186/s13662-018-1613-z.

[51]

D. Mozyrska, Multiparameter fractional difference linear control systems, Discrete Dyn. Nat. Soc., 2014 (2014), 8pp. doi: 10.1155/2014/183782.

[52]

D. Mozyrska and E. Girejko, Overview of fractional $h$-difference operators, in Advances in Harmonic Analysis and Operator Theory, Oper. Theory Adv. Appl., 229, Birkhäuser/Springer Basel AG, Basel, 2013, 253–268. doi: 10.1007/978-3-0348-0516-2_14.

[53]

D. Mozyrska, E. Girejko and M. Wyrwas, Comparison of $h$-difference fractional operators, in Advances in the Theory and Applications of Non-Integer Order Systems, Lect. Notes Electr. Eng., 257, Springer, Cham, 2013, 191–197. doi: 10.1007/978-3-319-00933-9_17.

[54]

A. Nagai, An integrable mapping with fractional difference, J. Phy. Soc. Japan, 72 (2003), 2181-2183.  doi: 10.1143/JPSJ.72.2181.

[55] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. 
[56]

F. Sabzikar, M. M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys. . 293 (2015), 14–28. doi: 10.1016/j. jcp. 2014.04.024.

[57]

M. R. SagayarajA. G. M. Selvam and M. P. Loganathan, On the oscillation of nonlinear fractional difference equations, Math. Aeterna, 4 (2014), 91-99. 

[58]

G. Sales TeodoroJ. A. Tenreiro Machado and E. Capelas de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195-208.  doi: 10.1016/j.jcp.2019.03.008.

[59]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Frational Integrals and Derivatives. Theory and Applications, Science and Technica, Minsk, 1987.

[60]

A. G. M. SelvamM. R. Sagayaraj and M. P. Loganathan, Oscillatory behavior of a class of fractional difference equations with damping, Inter. J. Appl. Math. Research, 3 (2014), 220-224.  doi: 10.14419/ijamr.v3i3.2624.

[61]

I. SuwanS. Owies and T. Abdeljawad, Fractional $h$-differences with exponential kernels and their monotonicity properties, Math. Methods Appl. Sci., 44 (2021), 8432-8446.  doi: 10.1002/mma.6213.

[62]

I. Suwan, S. Owies, M. Abussa and T. Abdeljawad, Monotonicity analysis of fractional proportional differences, Discrete Dyn. Nat. Soc., 2020 (2020), 11pp. doi: 10.1155/2020/4867927.

[63]

T. Yalçin Uzun, Oscillatory behavior of nonlinear Hilfer fractional difference equations, Adv. Difference Equ., 2021 (2021), 11pp. doi: 10.1186/s13662-021-03343-7.

show all references

References:
[1]

T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal, 2012 (2012), 13pp. doi: 10.1155/2012/406757.

[2]

T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), 12pp. doi: 10.1155/2013/406910.

[3]

T. Abdeljawad, Different type kernel $h$-fractional differences and their fractional $h$-sums, Chaos Solitons Fractals, 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.

[4]

T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Difference Equ., 2013 (2013), 16pp. doi: 10.1186/1687-1847-2013-36.

[5]

T. Abdeljawad, Fractional difference operators with discrete generalized Mittag-Leffler kernels, Chaos Solitons Fractals, 126 (2019), 315-324.  doi: 10.1016/j.chaos.2019.06.012.

[6]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.

[7]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Difference Equ., 2016 (2016), 18pp. doi: 10.1186/s13662-016-0949-5.

[8]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.

[9]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J Comput. Anal. Appl., 13 (2011), 574-582. 

[10]

T. Abdeljawad, D. Baleanu, F. Jarad and R. P. Agarwal, Fractional sums and differences with binomial coefficients, Discrete Dyn. Nat. Soc., 2013 (2013), 6pp. doi: 10.1155/2013/104173.

[11]

T. Abdeljawad, S. Banerjee and G. -C. Wu, Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption, Optik, 218 (2020). doi: 10.1016/j. ijleo. 2019.163698.

[12]

T. AbdeljawadF. Jarad and J. Alzabut, Fractional proportional differences with memory, Eur. Phys. J. Special Topics, 226 (2017), 3333-3354.  doi: 10.1140/epjst/e2018-00053-5.

[13]

J. O. Alzabut and T. Abdeljawad, Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl., 5 (2014), 177-187. 

[14]

G. A. Anastassiou, Discrete fractional caculus and inequalities, preprint, 2009, arXiv: 0911.3370.

[15]

G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling, 51 (2010), 562-571.  doi: 10.1016/j.mcm.2009.11.006.

[16]

G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Modelling, 52 (2010), 556-566.  doi: 10.1016/j.mcm.2010.03.055.

[17]

G. A. Anastassiou, Right nabla discrete fractional calculus, Int. J. Difference Equ., 6 (2011), 91-104. 

[18]

D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137. 

[19]

F. M. Atıcı and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176. 

[20]

F. M. Atıcı and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., (2009), 1–12. doi: 10.14232/ejqtde. 2009.4.3.

[21]

Z. Bai and R. Xu, The asymptotic behavior of solutions for a class of nonlinear fractional difference equations with damping term, Discrete Dyn. Nat. Soc., 2018 (2018), 11pp. doi: 10.1155/2018/5232147.

[22]

P. Baliarsingh, On a fractional difference operator, Alex. Eng. J., 55 (2016), 1811-1816.  doi: 10.1016/j.aej.2016.03.037.

[23]

P. Baliarsingh and L. Nayak, A note on fractional difference operators, Alex. Eng. J., 57 (2018), 1051-1054.  doi: 10.1016/j.aej.2017.02.022.

[24]

N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Discrete-time fractional variational problems, Signal Process., 91 (2011), 513-524.  doi: 10.1016/j.sigpro.2010.05.001.

[25]

N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., 29 (2011), 417-437.  doi: 10.3934/dcds.2011.29.417.

[26]

N. R. O. Bastos and D. F. M. Torres, Combined delta-nabla sum operator in discrete fractional calculus, Commun. Frac. Calc., 1 (2010), 41-47. 

[27]

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-0-8176-8230-9.

[28]

S. Chapman, On non-integral orders of summability of series and integrals, Proc. London Math. Soc. (2), 9 (1911), 369-409.  doi: 10.1112/plms/s2-9.1.369.

[29]

F. Chen, X. Luo and Y. Zhou, Existence results for nonlinear fractional difference equation, Adv. Difference Equ., 2011 (2011), 12pp. doi: 10.1155/2011/713201.

[30]

G. V. S. R. Deekshitulu and J. J. Mohan, Fractional difference inequalities, Commun. Appl. Anal., 14 (2010), 89-97. 

[31]

G. V. S. R. Deekshitulu and J. J. Mohan, Some new fractional difference inequalities, in Mathematical Modelling and Scientific Computation, Commun. Comput. Inf. Sci., 283, Springer, Heidelberg, 2012, 403–412. doi: 10.1007/978-3-642-28926-2_44.

[32]

G. V. S. R. Deekshitulu and J. J. Mohan, Some new fractional difference inequalities of Gronwall-Bellman type, Math. Sci., 6 (2012), 9pp.

[33]

E. C. de Oliveira and J. A. Tenreiro Machado, A review of definitions for fractional derivatives and integrals, Math. Probl. Eng., 2014 (2014), 6pp. doi: 10.1155/2014/238459.

[34]

J. B. Diaz and T. J. Osler, Differences of fractional order, Math. Comp., 28 (1974), 185-202.  doi: 10.1090/S0025-5718-1974-0346352-5.

[35]

R. K. GhazianiW. Govaerts and C. Sonck, Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response, Nonlinear. Anal. Real World Appl., 13 (2012), 1451-1465.  doi: 10.1016/j.nonrwa.2011.11.009.

[36]

H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp., 50 (1988), 513-529.  doi: 10.1090/S0025-5718-1988-0929549-2.

[37]

S. S. Haider and M. ur Rehman, On substantial fractional difference operator, Adv. Difference Equ., 2020 (2020), 18pp. doi: 10.1186/s13662-020-02594-0.

[38]

S. S. Haider, M. ur Rehman and T. Abdejawad, On Hilfer fractional difference operator, Adv. Difference Equ., 2020 (2020), 20pp. doi: 10.1186/s13662-020-02576-2.

[39]

R. Hirota, Lectures on difference equations, Science-sha, Japanese, 2000.

[40]

M. Holm, The Theory of Discrete Fractional Calculus Development and Application, Ph. D thesis, University of Nebraska in Lincoln, 2011.

[41]

J. R. M. Hosking, Fractional differencing, Biometrika, 68 (1981), 165-176.  doi: 10.1093/biomet/68.1.165.

[42]

S. C. Jun, A note on fractional differences based on a linear combination between forward and backward differences, Comput. Math. Appl., 41 (2001), 373-378.  doi: 10.1016/S0898-1221(00)00280-7.

[43]

T. Kaczorek, Positive 1D and 2D Systems, Communications and Control Engineering, Springer, London, 2002. doi: 10.1007/978-1-4471-0221-2.

[44]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.

[45]

B. Kuttner, On differences of fractional order, Proc. London Math. Soc. (3), 7 (1957), 453-466.  doi: 10.1112/plms/s3-7.1.453.

[46]

W. N. Li, Oscillation results for certain forced fractional difference equations with damping term, Adv. Difference Equ., 2016 (2016), 9pp. doi: 10.1186/s13662-016-0798-2.

[47]

K. S. Miller and B. Ross, Fractional difference calculus, in Univalent Functions, Fractional Calculus, and Their Applications (Kōriyama, 1988), Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989, 139–152.

[48]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.

[49]

P. O. Mohammeda, T. Abdeljawadb and F. K. Hamasalh, Discrete Prabhakar fractional difference and sum operators, Chaos Solitons Fractals, 150 (2021), 11pp. doi: 10.1016/j. chaos. 2021.111182.

[50]

A. Mouaouine, A. Boukhouima, K. Hattaf and N. Yousfi, A fractional order SIR epidemic model with nonlinear incidence rate, Adv. Difference Equ., 2018 (2018), 9pp. doi: 10.1186/s13662-018-1613-z.

[51]

D. Mozyrska, Multiparameter fractional difference linear control systems, Discrete Dyn. Nat. Soc., 2014 (2014), 8pp. doi: 10.1155/2014/183782.

[52]

D. Mozyrska and E. Girejko, Overview of fractional $h$-difference operators, in Advances in Harmonic Analysis and Operator Theory, Oper. Theory Adv. Appl., 229, Birkhäuser/Springer Basel AG, Basel, 2013, 253–268. doi: 10.1007/978-3-0348-0516-2_14.

[53]

D. Mozyrska, E. Girejko and M. Wyrwas, Comparison of $h$-difference fractional operators, in Advances in the Theory and Applications of Non-Integer Order Systems, Lect. Notes Electr. Eng., 257, Springer, Cham, 2013, 191–197. doi: 10.1007/978-3-319-00933-9_17.

[54]

A. Nagai, An integrable mapping with fractional difference, J. Phy. Soc. Japan, 72 (2003), 2181-2183.  doi: 10.1143/JPSJ.72.2181.

[55] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999. 
[56]

F. Sabzikar, M. M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys. . 293 (2015), 14–28. doi: 10.1016/j. jcp. 2014.04.024.

[57]

M. R. SagayarajA. G. M. Selvam and M. P. Loganathan, On the oscillation of nonlinear fractional difference equations, Math. Aeterna, 4 (2014), 91-99. 

[58]

G. Sales TeodoroJ. A. Tenreiro Machado and E. Capelas de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195-208.  doi: 10.1016/j.jcp.2019.03.008.

[59]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Frational Integrals and Derivatives. Theory and Applications, Science and Technica, Minsk, 1987.

[60]

A. G. M. SelvamM. R. Sagayaraj and M. P. Loganathan, Oscillatory behavior of a class of fractional difference equations with damping, Inter. J. Appl. Math. Research, 3 (2014), 220-224.  doi: 10.14419/ijamr.v3i3.2624.

[61]

I. SuwanS. Owies and T. Abdeljawad, Fractional $h$-differences with exponential kernels and their monotonicity properties, Math. Methods Appl. Sci., 44 (2021), 8432-8446.  doi: 10.1002/mma.6213.

[62]

I. Suwan, S. Owies, M. Abussa and T. Abdeljawad, Monotonicity analysis of fractional proportional differences, Discrete Dyn. Nat. Soc., 2020 (2020), 11pp. doi: 10.1155/2020/4867927.

[63]

T. Yalçin Uzun, Oscillatory behavior of nonlinear Hilfer fractional difference equations, Adv. Difference Equ., 2021 (2021), 11pp. doi: 10.1186/s13662-021-03343-7.

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