Some simple representations of the generalized mittag-leffler functions

In this paper we shall derive, in a simple fashion, formulas that express various classes of generalized Mittag-Leffier functions E αβ(x) in closed form, even though some of the results are known. In Section 1 we recall a few useful identities involving Mittag-Leffler functions that will be useful in our work. In Section 2 we derive a simple basic formula that expresses E αβ(x) in terms of the incomplete gamma function when α is a rational number and β is arbitrary. For example, various useful special cases, when the parameters α and β are both rational, are exhibited. In particular the case when α and β are both integers is treated in more detail in Section 3. The representations in this case may be written in terms of elementary functions.