A-stability of near-diagonal exponential Padé approximants ($n-2\le m\le n$)

For nonnegative degrees $m,n$ with $n-2\le m\le n$, the $[m/n]$ Padé approximant $R$ to the exponential function has no poles in the left half-plane and satisfies $|R(z)|\le 1$ for $\mathrm{Re}z\le 0$. Consequently a numerical method having that stability function is A-stable. In particular, the near-diagonal choice $[s-1/s]$ gives the Radau IIA method stability function, while $[s/s]$ corresponds to the Gauss collocation method. The theorem concerns these exponential approximants, not arbitrary rational approximations with the same degrees.