On the Incomplete Gamma and Biparametric Mittag-Leffler Functions

This paper presents new developments concerning the biparametric Mittag-Leffler function, Eα,β(z). First, numerous results are derived when the primary parameter, α, is set equal to unity. Many of these results are related to the incomplete gamma function, Γ(a,z), or special cases of it, such as the error function. It is also found that the parameter a is related to the secondary parameter of the biparametric Mittag-Leffler function, β. More results are derived by considering other values, both fixed and algebraic, of the primary parameter. In particular, it is shown that, for rational values, the biparametric Mittag-Leffler function can be expressed as a finite sum of α=1 functions or incomplete gamma functions. Then, the asymptotic forms of the incomplete gamma function are made exact for z situated over the entire principal branch by the application of the regularization techniques, Borel summation and Mellin–Barnes regularization. Introducing the asymptotic forms for the incomplete gamma function into the newly derived results of the biparametric Mittag-Leffler function results in asymptotic forms that give exact values of the function, which, where possible, are checked with values from the MittagLefflerE instruction in Mathematica.