Endpoint-constrained rational approximation to a power ($r(x)=x/[k+(1-k)x]$)

For $0<k\le 1$, impose $r(0)=0$, $r(1)=1$, and $r^{\prime }(1)=k$ within the rational family $ax/(1+bx)$. Matching the first two Taylor coefficients at $x=1$ gives the displayed formula. It differs from the standard degree-$(1,1)$ Padé approximant to $x^k$ about one, which is $[(1-k)+(1+k)x]/[(1+k)+(1-k)x]$ and matches one further Taylor coefficient. At $k=0$, the constrained expression is interpreted by its removable constant limit.