Combining arbitrary order global Pad\'e approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost

TL;DR

This paper enhances the accuracy of Mittag-Leffler function approximations by combining global Padé approximation with its addition formula, providing practical implementations in Mathematica and C++ for improved computational results.

Contribution

It introduces a novel combination of Padé approximation and addition formula for Mittag-Leffler functions, with detailed implementation strategies.

Findings

Significantly higher accuracy with combined approximation methods

Effective computational solutions in Mathematica and C++

Comparison with contour integral methods demonstrates improved performance

Abstract

The combination of the global Pad\'e approximation of the Mittag-Leffler function with its addition formula for the case $\alpha<1$ yields significantly higher accuracy results for a given arbitrary order $n$. We present a solution in terms of a Mathematica notebook to determine the general structure of the system of linear equations to be solved, followed by an implementation as a {\tt{C++}} program using the {\tt{Eigen}} template library for linear algebra. For a comparison with contour integral solutions we present an implementation as a {\tt{C++}} program using the {\tt{boost}} library's quadrature package employing the Gauss-Kronrod-method.