Nondiagonal quadratic Hermite-Padé approximation to the exponential function

Approximation to the exponential function of the form Emns(z) := Pn(z)e−2z + Qm(z)e−z + Rs(z) = O(zm+n+s+2) where Pn, Qm and Rs are polynomials of degree at most n,m and s, respectively, and Pn has leading coefficient 1 is considered. Explicit formulas for the polynomials Pn, Qm and Rs are obtained by using some identities involving hypergeometric series in the case of Rs. An exact asymptotic expression as m + n → ∞ is obtained for the error function Emns and it is also shown that these Hermite-Padé Type I polynomials can be used to asymptotically minimize expressions of the above form on the unit disk.