Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function (Formula presented.) is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When (Formula presented.), the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for (Formula presented.) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for (Formula presented.), (Formula presented.), with (Formula presented.). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.