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

Abstract

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.