Rational Approximation for Oscillatory Mittag-Leffler Function

Mittag-Leffler function $E_{\alpha}$ and its generalizations are indispensable in fractional calculus. When $1\lt\alpha\lt2$, its behavior is so diverse since it could be nonmonotone with oscillatory profile and varying odd number of real roots. Due to these properties, it is a challenging task to approximate this function in this range of $\alpha$. In this paper, we develop a rational approximation for $E_{\alpha}(-z),z\gt0$, that captures the oscillatory behavior and the roots over extended intervals. The approximation is based on decomposing $E_{\alpha}$ into a combination of a two-parameter rootless Mittag-Leffler function and a polynomial. This type of approximations provides efficient implementations when numerically solving fractional oscillation equations.