Global Padé approximations of the generalized Mittag-Leffler function and its inverse

Abstract

This paper proposes a global Padé approximation of the generalized Mittag-Leffler function $E_{α,β}(-x)$ 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 $E_{α,β}(-x)$, we work out the global Padé approximation [1/2] for the particular cases $\{0<α<1, β>α\}$, $\{0<α=β<1\}$, and $\{α=1, β>1\}$, respectively. Moreover, these approximations are inverted to yield a global Padé approximation of the inverse generalized Mittag-Leffler function $-L_{α,β}(x)$ with $x\in(0,1/Γ(β)]$. We also provide several examples with selected values $α$ and $β$ 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.