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**On right fractional calculus.**
*(English)*
Zbl 1198.26006

Summary: Here are presented fractional Taylor type formulae with fractional integral remainder and fractional differential formulae, regarding the right Caputo fractional derivative, the right generalized fractional derivative of J. A. Canavati type [Nieuw Arch. Wiskd., IV. Ser. 5, 53–75 (1987; Zbl 0649.46026)] and their corresponding right fractional integrals.Then are given representation formulae of functions as fractional integrals of their above fractional derivatives, as well as of their right and left Weyl fractional derivatives.At the end, we mention some far reaching implications of our theory to mathematical analysis computational methods.Also we compare the right Caputo fractional derivative to right Riemann-Liouville fractional derivative.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

### MSC:

26A33 | Fractional derivatives and integrals |

### Citations:

Zbl 0649.46026
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\textit{G. A. Anastassiou}, Chaos Solitons Fractals 42, No. 1, 365--376 (2009; Zbl 1198.26006)

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### References:

[1] | Anastassiou, G., Quantitative approximations (2001), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, New York · Zbl 0969.41001 |

[2] | Canavati, J. A., The Riemann-Liouville integral, Nieuw Archief Voor Wiskunde, 5, 1, 53-75 (1987) · Zbl 0649.46026 |

[3] | Kai Diethelm, Fractional differential equations. Available from: http://www.tu-bs.de/ diethelm/lehre/f-dgl02/fde-skript.ps.gz; Kai Diethelm, Fractional differential equations. Available from: http://www.tu-bs.de/ diethelm/lehre/f-dgl02/fde-skript.ps.gz · Zbl 1170.34022 |

[4] | El-Sayed, A. M.A.; Gaber, M., On the finite Caputo and finite Riesz derivatives, Electron J Theor Phys, 3, 12, 81-95 (2006) · Zbl 1236.35003 |

[5] | Frederico, G. S.; Torres, D. F.M., Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem, Int Math Forum, 3, 10, 479-493 (2008) · Zbl 1154.49016 |

[6] | Gorenflo R, Mainardi F, Essentials of fractional calculus. Maphysto Center; 2000. Available from: http://www.maphysto.dk/oldpages/events/LevyCAC2000/MainardiNotes/fm2k0a.ps; Gorenflo R, Mainardi F, Essentials of fractional calculus. Maphysto Center; 2000. Available from: http://www.maphysto.dk/oldpages/events/LevyCAC2000/MainardiNotes/fm2k0a.ps |

[7] | Miller, K. S., The Weyl fractional calculus, (Ross, B., Fractional calculus and its applications. Fractional calculus and its applications, Lecture Notes in Mathematics, vol. 457 (1975), Springer-Verlag: Springer-Verlag Berlin, New-York), 80-89 · Zbl 0305.44004 |

[8] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives, theory and applications (1993), Gordon and Breach: Gordon and Breach Amsterdam, [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)] · Zbl 0818.26003 |

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