Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse

Abstract

Abstract This paper proposes a global Padé approximation of the generalized Mittag-Leffler function Eα,β(−x) with x ∈ [0,+∞). 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 Eα,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 −Lα,β(x) with x ∈ (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.