Subresultants of $(x-alpha)^m$ and $(x-beta)^n$, Jacobi polynomials and complexity

In an earlier article together with Carlos DAndrea [BDKSV2017], we described explicit expressions for the coefficients of the order-$d$ polynomial subresultant of $(x-alpha)^m$ and $(x-beta)^n $ with respect to Bernsteins set of polynomials ${(x-alpha)^j(x-beta)^{d-j}, , 0le jle d}$, for $0le d<min{m, n}$. The current paper further develops the study of these structured polynomials and shows that the coefficients of the subresultants of $(x-alpha)^m$ and $(x-beta)^n$ with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials. The result is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are scalar multiples of Jacobi polynomials up to an affine change of variables.