@PietroMajer: Rewrite $r_n$ and $s_n$ (e.g., from the power basis) to the Bernstein form of degree exactly $n+1$ (see the section "Background"), and then take their coefficients in the Bernstein form. These coefficients (labeled $a_k$ in the section "Background") are the Bernstein coefficients (of degree $n+1$) of $r_n$ and $s_n$.

@PietroMajer: $r_n$ and $s_n$ -- not their Bernstein polynomials $B_m r_n$ and $B_m s_n$, which are different in general -- must be polynomials and must have non-negative Bernstein coefficients once they are rewritten to polynomials in Bernstein form of degree exactly $n+1$.