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Abdeljawad, Thabet; Baleanu, Dumitru; Jarad, Fahd; Agarwal, Ravi P.

Fractional sums and differences with binomial coefficients. (English) Zbl 1264.26006

Discrete Dyn. Nat. Soc. 2013, Article ID 104173, 6 p. (2013).
Summary: In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

Cited in 27 Documents

MSC:

26A33 Fractional derivatives and integrals
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\textit{T. Abdeljawad} et al., Discrete Dyn. Nat. Soc. 2013, Article ID 104173, 6 p. (2013; Zbl 1264.26006)
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References:

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