Global Padé approximations of the generalized Mittag-Leffler function and its inverse (Q898908)

The authors develop global Padé approximations for the generalized Mittag-Leffler function \(E_{\alpha,\beta}(-x)\) for the cases \(\{0<\alpha <1, \beta > \alpha \}\), \(\{0<\alpha = \beta <1 \}\), and \(\{\alpha =1, \beta > 1 \}\), respectively with \(x\in [0,+\infty).\) Furthermore, the inverse generalized Mittag-Leffler function \(-L_{\alpha, \beta}(x)\) was also studied and its uniform approximations derived. Applications of the established results to ordinary and partial time-fractional differential equations in the sense of Riemann-Liouville are given and discussed.