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-Leffler functions E-alpha beta(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-alpha beta(x) in terms of the incomplete gamma function when alpha is a rational number and beta is arbitrary. For example, various useful special cases, when the parameters alpha and beta are both rational, are exhibited. In particular the case when alpha and beta are both integers is treated in more detail in Section 3. The representations in this case may be written in terms of elementary functions.