Global Pad\'e approximations of the generalized Mittag-Leffler function and its inverse

This paper proposes a global Pad\'{e} approximation of the generalized Mittag-Leffler function $E_{\alpha,\beta}(-x)$ with $x\in[0,+\infty)$. This uniform approximation can account for both the Taylor series for small arguments and asymptotic series for large arguments. Based on the complete monotonicity of the function $E_{\alpha,\beta}(-x)$, we work out the global Pad\'{e} approximation [1/2] for the particular cases $\{0<\alpha<1, \beta>\alpha\}$, $\{0<\alpha=\beta<1\}$, and $\{\alpha=1, \beta>1\}$, respectively. Moreover, these approximations are inverted to yield a global Pad\'{e} approximation of the inverse generalized Mittag-Leffler function $-L_{\alpha,\beta}(x)$ with $x\in(0,1/\Gamma(\beta)]$. We also provide several examples with selected values $\alpha$ and $\beta$ to compute the relative error from the approximations. Finally, we point out the possible applications using our established approximations in the ordinary and partial time-fractional differential equations in the sense of Riemann-Liouville.