Quadratic Hermite-Padé approximation to the exponential function

Abstract

Approximation to exp of the form

wherep _m,q _m, andr _m are polynomials of degree at mostm andp _m has lead coefficient 1 is considered. Exact asymptotics and explicit formulas are obtained for the sequences {ℰ _m}, {p _m}, {q _m}, and {r _m}. It is observed that the above sequences all satisfy the simple four-term recursion:

$$\begin{array}{*{20}c} {T_{m + 3} = \frac{1}{{3m + 4}}[( - 6m - 14)z^3 T_m } \\ { + (9m + 15)(z^2 + (3m + 4)(3m + 7))T_{m + 1} + 3zT_{m + 2} ].} \\ \end{array} $$

It is also observed that these generalized Padé-type approximations can be used to asymptotically minimize expressions of the above form on the unit disk.