In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.

翻译：在本文中,我们引入了一套合理近似的组合,即用于明确解决某些线性差异方程式以及逐渐演变的整合办法的美元功能的对等性,并介绍了这一组合的近似值的推算和特性,适用于天平和矩阵参数。此外,我们表明,这些近似值所计算的矩阵功能显示出与现有最佳理论界限相当的衰减性能。数字实例突出显示了拟议中的合理近似值的好处(r.t.~经典的泰勒多面体和其他理性功能)。