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Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

Author

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  • Alexander Apelblat

    (Department of Chemical Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel)

Abstract

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

Suggested Citation

  • Alexander Apelblat, 2020. "Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach," Mathematics, MDPI, vol. 8(5), pages 1-22, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:657-:d:350627
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    References listed on IDEAS

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    1. Sandev, Trifce & Tomovski, Živorad & Dubbeldam, Johan L.A., 2011. "Generalized Langevin equation with a three parameter Mittag-Leffler noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3627-3636.
    2. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
    3. Biyajima, Minoru & Mizoguchi, Takuya & Suzuki, Naomichi, 2015. "A new blackbody radiation law based on fractional calculus and its application to NASA COBE data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 440(C), pages 129-138.
    4. Agahi, Hamzeh & Alipour, Mohsen, 2019. "Mittag-Leffler-Gaussian distribution: Theory and application to real data," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 227-235.
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    Cited by:

    1. Juan Luis González-Santander & Fernando Sánchez Lasheras, 2022. "Finite and Infinite Hypergeometric Sums Involving the Digamma Function," Mathematics, MDPI, vol. 10(16), pages 1-12, August.
    2. Fabio Vanni & David Lambert, 2024. "Aging Renewal Point Processes and Exchangeability of Event Times," Mathematics, MDPI, vol. 12(10), pages 1-26, May.
    3. Juan Luis González-Santander & Fernando Sánchez Lasheras, 2023. "Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions," Mathematics, MDPI, vol. 11(8), pages 1-16, April.
    4. Alexander Apelblat & Juan Luis González-Santander, 2021. "The Integral Mittag-Leffler, Whittaker and Wright Functions," Mathematics, MDPI, vol. 9(24), pages 1-34, December.

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