Padé–biroot correspondence for square-root approximations

Let $P_{[p/q]}^{(c)}(x)$ denote the $[p/q]$ Padé approximant associated with the square-root expansion at $c$, and let $\beta_m^2(x,\sqrt{c})$ be the square biroot approximation. Set

$$m=p+q+1,$$

with $q\in\{p,p-1\}$. Padé–biroot correspondence. The approximant satisfies

$$P_{[p/q]}^{(c)}(x)\sim\sqrt{x}=\beta_m^2(x,\sqrt{c}).$$

The conjecture proposes that the binomial-coefficient construction underlying the biroot method reproduces the relevant Padé approximants, including the observed correspondence between Newton iterations and Padé approximations for square roots.