Rational approximation of $x^n$

Let $E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in approximation of $x^n$ on $[0,1]$ by rational functions of type $(k,k)$ with $k<n$. We show that in an appropriate limit $E_{kk}^{(n)}\sim 2H^{k+1\mathrm{/}2}$ independently of $n$, where $H\approx 1\mathrm{/}9.28903$ is Halphen’s constant. This is the same formula as for minimax approximation of $e^x$ on $(-\mathrm{\infty },0]$.