Communications of the
Korean Mathematical Society CKMS

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.

Keywords: gamma function, beta function, extended beta function, Mittag-Leffler function, extended Mittag-Leffler functions, Fox-Wright function, generalized hypergeometric function, Mellin transform, Riemann-Liouville fractional derivative, extended Riemann-Liouvi

MSC numbers: 33B20, 33C20, 33C45, 33C60, 33B15, 33C05, 26A33