On extended $ k $-generalized Mittag-Leffler function and its properties

Abstract

In this current paper, we are using the concept of extension of the beta function to define an extended $ k $-generalized Mittag-Leffler function (GMLf) $ E_{k, l, m}^{\rho, \sigma;c}(x;p) $. There are four sections included in this paper containing some properties of the above-described function, like derivatives, integral representation, and integral transform. The establishment of some recurrence relations has also been done. We also derive the extended $ k $-GMLf from the extended $ k $-Riemann-Liouville (R-L) fractional derivative of generalized MLf. Numerous former results studied by many researchers can also be derived as special cases of our results.