On the Incomplete Gamma and Biparametric Mittag-Leffler Functions

This paper presents new developments concerning the biparametric Mittag-Leffler function, $E_{\alpha,\beta}(z)$. First, numerous results are derived when the primary parameter, $\alpha$, is set equal to unity. Many of the these results are related to the incomplete gamma function, $\Gamma(a,z)$, 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, $\beta$. 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 $\alpha=1$ 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.