**Hari M. Srivastava 1,2,\*, Arran Fernandez 3,4 and Dumitru Baleanu 5,6**


Received: 30 March 2019; Accepted: 20 May 2019; Published: 28 May 2019

**Abstract:** We consider the well-known Mittag–Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag–Leffler function as a fractional derivative of the two-parameter Mittag–Leffler function, which is in turn a fractional integral of the one-parameter Mittag–Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag–Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.

**Keywords:** fractional integrals; fractional derivatives; Mittag–Leffler functions

**MSC:** 26A33; 33E12
