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

Alexander Apelblat
Additional contact information
Alexander Apelblat: Department of Chemical Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

Mathematics, 2020, vol. 8, issue 5, 1-22

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.

Keywords: derivatives with respect to parameters; Mittag-Leffler functions; Laplace transform approach; infinite power series; integral representations; convolution integrals; quotients of digamma and gamma functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/5/657/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/5/657/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:5:p:657-:d:350627

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().