• Communications of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.549-560
• /
• 2018

Since Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903, due to its usefulness and diverse applications, a variety and large number of its extensions (and generalizations) and variants have been presented and investigated. In this sequel, we aim to introduce a new extension of the Mittag-Leffler function by using a known extended beta function. Then we investigate ceratin useful properties and formulas associated with the extended Mittag-Leffler function such as integral representation, Mellin transform, recurrence relation, and derivative formulas. We also introduce an extended Riemann-Liouville fractional derivative to present a fractional derivative formula for a known extended Mittag-Leffler function, the result of which is expressed in terms of the new extended Mittag-Leffler functions.

• Communications of the Korean Mathematical Society
• /
• v.35 no.1
• /
• pp.161-184
• /
• 2020

In this article, we establish some new weighted Hardy-type inequalities involving some variants of extended Riemann-Liouville fractional derivative operators, using convex and increasing functions. As special cases of the main results, we obtain the results of [18,19]. We also prove the boundedness of the k-fractional integral operator on L_p[a, b].

• Communications of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.507-522
• /
• 2019

The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of beta function recently defined by Shadab et al. [19]. Moreover, we establish some results related to the newly defined modified fractional derivative operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

• Honam Mathematical Journal
• /
• v.46 no.3
• /
• pp.335-348
• /
• 2024

This paper deals with a new extension of the generalized Bessel-Maitland function (EGBMF) associated with the beta function. We evaluated integral representations, recurrence relation and integral transforms such as Mellin transform, Laplace transform, Euler transform, K-transform and Whittaker transform. Furthermore, the Riemann-Liouville fractional integrals are also discussed.