TY - JOUR

T1 - Differentiation of the mittag-leffier functions with respect to parameters in the laplace transform approach

AU - Apelblat, Alexander

N1 - Funding Information:
This research received no external funding. I wish to express my profound gratitude to Francesco Mainardi, Department of Physics and Astronomy, Bologna University, Bologna, Italy, for his help, advice, and kind encouragement over the years. He was the first person to show me the importance of the Mittag-Leer functions. I am also grateful to Yuri A. Brychkov from the Computing Center of the Russian Academy of Sciences, Moscow, Russia, and Victor Adamchik from the Computer Science Department, University of Southern California, Los Angeles, USA, for simplifying and verifying several series fromMATHEMATICA in this work. I thank Juan Luis Gonzales-Santander Martinez, Department of Mathematics, Universidad de Oviedo, Oviedo, Spain, for explaining MATHEMATICA and producing more elegant and efficient programs from my original programs. Finally, I thank the referees for their constructive comments, which helped improve this work considerably.
Publisher Copyright:
© 2020 by the authors.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - 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.

AB - 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.

KW - Convolution integrals

KW - Derivatives with respect to parameters

KW - Infinite power series

KW - Integral representations

KW - Laplace transform approach

KW - Mittag-Leffler functions

KW - Quotients of digamma and gamma functions

UR - http://www.scopus.com/inward/record.url?scp=85085568900&partnerID=8YFLogxK

U2 - 10.3390/MATH8050657

DO - 10.3390/MATH8050657

M3 - Article

AN - SCOPUS:85085568900

VL - 8

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 5

M1 - 657

ER -