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 (a = 1, ß > 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/Г(β)]. We also provide several examples with selected values a and ß 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.
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Zeng, C., Chen, Y.Q. Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse. FCAA 18, 1492–1506 (2015). https://doi.org/10.1515/fca-2015-0086
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DOI: https://doi.org/10.1515/fca-2015-0086