Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential

We study uniform approximation of $\exp(tz)$, $t>0$, on $(-\infty,0]$ by rational functions $P_m(z)/(q_m-z)^m$ with a prescribed repeated pole $q_m>0$. A Möbius transformation reduces the problem to polynomial approximation of $F_λ(x)=\exp\!\left(-λ\frac{1-x}{1+x}\right)$ on $[-1,1]$, where $λ=tq_m$