A Class of Extended Mittag–Leffler Functions and Their Properties Related to Integral Transforms and Fractional Calculus

Abstract

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out.

Keywords

• extended Beta function
• extended hypergeometric functions
• extended confluent hypergeometric function
• Mittag–Leffler function
• generalized Mittag–Leffler function
• Laplace transform
• Mellin transform
• Euler-Beta transforms
• Wright hypergeometric function
• Fox H-function
• fractional calculus operators