Affine Scaling of Jacobi Zeros: Sharp Orderings Beyond Gautschi's Conjectures
We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(α,β)}$, $α,β>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|β|\leq1/2$. For the spectral scale $n+(α+β+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.