General nondiagonal cubic Hermite-Padé approximation to the exponential function (Q2477920)

So called cubic Hermite-Padé approximants for \(\exp(-x)\) can be generated form the polynomials \(P_k\) of degree \(\delta_k\) at most that satisfy \(\sum_{k=0}^3 P_k(x)\exp(-kx)=O(x^q)\) with \(q=\sum_{k=0}^3\delta_k+3\) and \(P_3\) monic. First it is shown how the polynomials \(P_k\) of the cubic approximation can be obtained from the polynomials of the quadratic approximation, which in turn can be obtained from the linear (i.e., ordinary) Padé approximation. The main objective of the paper is to give asymptotics for the \(O\)-term as \(\delta_3+\delta_2\to\infty\). The result is of the form \(c x^q \exp(px)(1+o(1))\) with \(q\) as above and \(c\) and \(p\) depending on all \(\delta_k\)'s. This holds uniformly in compact subsets of the complex plane. The asymptotic best uniform approximant in the unit disk is discussed in particular. Finally the differential equations satisfied by the polynomials \(P_k\) are given.