Nondiagonal quadratic Hermite-Padé approximation to the exponential function (Q1917934)

The author considers Hermite-Padé approximation to the exponential function of the form \[ E_{mns} (x) = P_n (x) e^{-2x} + Q_m (x) e^{-x} + R_s (x) = O(x^{n + m + s + 2}) \tag{*} \] as \(x \to 0\), where \(P_n\), \(Q_m\) and \(R_s\) are polynomials of degree at most \(n,m\) and \(s\), respectively, and \(P_n\) monic (Hermite-Padé Type I polynomials). Explicit formulas for these polynomials are obtained; in the case of \(R_s\), its coefficients have a connection with terminating hypergometric series evaluated at a certain point. Moreover, an exact asymptotic formula for the error function \(E_{mns} (x)\) holding locally uniformly on \(\mathbb{C}\) when \(m + n \to \infty\) is derived. Finally, the author shows that the problem of minimizing polynomial combinations of \(\{E^{- 2x}, e^{-x}, 1\}\) of type (*) on the unit disc is asymptotically solved by the Hermite-Padé Type I polynomials with shifted center of approximation.