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Keywords:
fractional difference; fractional sum; discrete Mittag-Leffler function
Summary:
The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution.
References:
[1] Atici, F. M., Eloe, P. W.: A transform method in discrete fractional calculus. Int. J. Difference Equ. 2 (2007), 165-176. MR 2493595
[2] Atici, F. M., Eloe, P. W.: Initial value problems in discrete fractional calculus. Proc. Amer. Math. Soc. 137 (2009), 981-989. DOI 10.1090/S0002-9939-08-09626-3 | MR 2457438 | Zbl 1166.39005
[3] Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, MA (2001). MR 1843232 | Zbl 0993.39010
[4] Čermák, J., Nechvátal, L.: On $(q,h)$-analogue of fractional calculus. J. Nonlinear Math. Phys. 17 (2010), 1-18. DOI 10.1142/S1402925110000593 | MR 2647460 | Zbl 1189.26006
[5] Gray, H. L., Zhang, N. F.: On a new definition of the fractional difference. Math. Comp. 50 (1988), 513-529. DOI 10.1090/S0025-5718-1988-0929549-2 | MR 0929549 | Zbl 0648.39002
[6] Miller, K. S., Ross, B.: Fractional Difference Calculus. Proc. Int. Symp. Unival. Funct., Frac. Calc. Appl., Koriyama, Japan, May 1988, 139-152; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989. MR 1199147 | Zbl 0693.39002
[7] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York (1993). MR 1219954 | Zbl 0789.26002
[8] Díaz, R., Teruel, C.: $q,k$-Generalized Gamma and Beta Functions. J. Nonlin. Math. Phys. 12 (2005), 118-134. DOI 10.2991/jnmp.2005.12.1.10 | MR 2122869 | Zbl 1075.33010
[9] Díaz, J. B., Osler, T. J.: Differences of fractional order. Math. Comp. 28 (1974), 185-202. DOI 10.2307/2005825 | MR 0346352
[10] Podlubný, I.: Fractional Differential Equations. Academic Press, San Diego (1999). MR 1658022
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