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Fractional Calculus of a Unified Mittag-Leffler Function

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The main aim of the paper is to introduce an operator in the space of Lebesgue measurable real or complex functions L(a, b). Some properties of the Riemann–Liouville fractional integrals and differential operators associated with the function E α,β,λ,μ,ρ,p γ,δ (cz; s, r) are studied and the integral representations are obtained. Some properties of a special case of this function are also studied by the means of fractional calculus.

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References

  1. 1.

    A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York (2010).

  2. 2.

    Applications of Fractional Calculus in Physics, Ed. R. Hilfer, World Scientific, Singapore, etc. (2000).

  3. 3.

    Fractional Time Evolution. Applications of Fractional Calculus in Physics, Ed. R. Hilfer, World Scientific, Singapore, etc. (2000).

  4. 4.

    A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,” Integral Transforms Spec. Funct., 15, 31–49 (2004).

  5. 5.

    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” North-Holland Math. Stud., Elsevier (North-Holland) Sci. Publ., Amsterdam, 204 (2006).

  6. 6.

    K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).

  7. 7.

    J. C. Prajapati, B. I. Dave, and B.V. Nathwani, “On a unification of generalized Mittag-Leffler function and family of Bessel functions,” Adv. Pure Math., 3, No. 1, 127–137 (2013).

  8. 8.

    E. D. Rainville, Special Functions, Macmillan Co., New York (1960).

  9. 9.

    T. O. Salim, “Some properties relating to the generalized Mittag-Leffler function,” Adv. Appl. Math. Anal., 4, No. 1, 21–30 (2009).

  10. 10.

    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Yverdon (Switzerland) (1993).

  11. 11.

    A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler functions and its properties,” J. Math. Anal. Appl., 337, 797–811 (2007).

  12. 12.

    H. M. Srivastava and R. K. Saxena, “Some Vottera type fractional integrodifferential equations with a multivariable confluent hypergeometric function as their kernel,” J. Integral Equat. Appl., 17, 199–217 (2005).

  13. 13.

    H. M. Srivastava and Z. Tomovski, “Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel,” Appl. Math. Comput., 211, No. 1, 198–210 (2009).

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Correspondence to J. C. Prajapati.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 8, pp. 1133–1145, August, 2014.

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Prajapati, J.C., Nathwani, B.V. Fractional Calculus of a Unified Mittag-Leffler Function. Ukr Math J 66, 1267–1280 (2015). https://doi.org/10.1007/s11253-015-1007-2

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Keywords

  • Differential Operator
  • Arbitrary Constant
  • Fractional Calculus
  • Fractional Differential Equation
  • Laplace Transformation