Fractional Sums and Differences with Binomial Coefficients
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 a...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2013/104173 |
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| _version_ | 1850229685334573056 |
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| author | Thabet Abdeljawad Dumitru Baleanu Fahd Jarad Ravi P. Agarwal |
| author_facet | Thabet Abdeljawad Dumitru Baleanu Fahd Jarad Ravi P. Agarwal |
| author_sort | Thabet Abdeljawad |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-21a30a619ee44948983c92fe5a2e272f |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-21a30a619ee44948983c92fe5a2e272f2025-08-20T02:04:07ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2013-01-01201310.1155/2013/104173104173Fractional Sums and Differences with Binomial CoefficientsThabet Abdeljawad0Dumitru Baleanu1Fahd Jarad2Ravi P. Agarwal3Department of Mathematics, Faculty of Art and Sciencs, Çankaya University, Balgat, 06530 Ankara, TurkeyDepartment of Mathematics, Faculty of Art and Sciencs, Çankaya University, Balgat, 06530 Ankara, TurkeyDepartment of Mathematics, Faculty of Art and Sciencs, Çankaya University, Balgat, 06530 Ankara, TurkeyDepartment of Mathematics, Texas A & M University, 700 University Boulevard, Kingsville, TX, USAIn 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.http://dx.doi.org/10.1155/2013/104173 |
| spellingShingle | Thabet Abdeljawad Dumitru Baleanu Fahd Jarad Ravi P. Agarwal Fractional Sums and Differences with Binomial Coefficients Discrete Dynamics in Nature and Society |
| title | Fractional Sums and Differences with Binomial Coefficients |
| title_full | Fractional Sums and Differences with Binomial Coefficients |
| title_fullStr | Fractional Sums and Differences with Binomial Coefficients |
| title_full_unstemmed | Fractional Sums and Differences with Binomial Coefficients |
| title_short | Fractional Sums and Differences with Binomial Coefficients |
| title_sort | fractional sums and differences with binomial coefficients |
| url | http://dx.doi.org/10.1155/2013/104173 |
| work_keys_str_mv | AT thabetabdeljawad fractionalsumsanddifferenceswithbinomialcoefficients AT dumitrubaleanu fractionalsumsanddifferenceswithbinomialcoefficients AT fahdjarad fractionalsumsanddifferenceswithbinomialcoefficients AT ravipagarwal fractionalsumsanddifferenceswithbinomialcoefficients |