Q-Mittag–Leffler Functions

Abstract

The classical Mittag–Leffler function plays an important role in fractional differential equations. In this chapter we mention in brief the q-analogues of the Mittag–Leffler functions defined by mathematicians. We pay attention to a pair of q-analogues of the Mittag–Leffler function that may be considered as a generalization of the q-exponential functions e q (z) and E q (z). We study their main properties and give a Mellin–Barnes integral representations and Hankel contour integral representation for them. As in the classical case we prove that the q-Mittag–Leffler functions are solutions of q-type Volterra integral equations. Finally, asymptotics of zeros of one of the pair of the q-Mittag–Leffler function will be given at the end of the chapter.