Global-Padé Approximation of the Three-Parameter Mittag-Leffler Function: Generalized Derivation and Numerical Implementation Issues

In this paper, we discuss a framework for efficient construction of rational approximations for the three-parameter Mittag-Leffler function (MLF) based on the global-Padé approximation technique. Obtaining the coefficients of these approximants is usually the most tedious part in their construction as it requires delicate matching to procure a linear system for these coefficients. The main focus of this paper is the derivation of a novel generalized system that can be used to obtain the coefficients without having to perform the matching task. In particular, we illustrate the use of the generalized system in constructing rational approximants of various degrees subject to choices of feasible parameters. Inequalities providing bounds on permissible choices of these parameters are also obtained. Numerical experiments are conducted to illustrate the accuracy and efficiency of the approximants.