Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant (Q1980748)

Let \(\mathbf{x}_n=(x_j)\in\mathbb{K}^n\) and \(\mathbf{a}_m=(a_j)\in\mathbb{K}^m\), where \(\mathbb{K}\) is \(\mathbb{R}\) or \(\mathbb{C}\). The generalized Vandermonde matrix \(G_{mn}(\mathbf{x}_n,\mathbf{a}_m)\in\mathbb{K}^{m\times n}\) with \((i,j)\)-th entry \(x_j^{a_i}\). The determinant of \(G_n(\mathbf{x}_n,\mathbf{a}_n)=G_{nn}(\mathbf{x}_n,\mathbf{a}_n)\) is the generalized Vandermonde determinant: \(g_n(\mathbf{x}_n,\mathbf{a}_n)=\det{G_n(\mathbf{x}_n,\mathbf{a}_n)}\). The Vandermonde matrix \(V_{mn}(\mathbf{x}_n)=G_{mn}(\mathbf{x}_n,\mathbf{b}_m)\) and the Vandermonde determinant \(v_n(\mathbf{x}_n)=g_n(\mathbf{x}_n,\mathbf{b}_n)\), where \(\mathbf{b}_m=(0,1,\dots,m-1)\). The authors first visualize \(v_3(\mathbf{x}_3)\) and also \(g_3(\mathbf{x}_3,\mathbf{a}_3)\) for certain choices of \(\mathbf{a}_3\). Second, they prove that a point \(\mathbf{x}_n\) on the unit sphere of \(\mathbb{R}^n\) is an extreme point of \(v_n\) if and only if \(x_1,\dots,x_n\) are distinct roots of the rescaled Hermite polynomial \[ P_n(x)=\frac{H_n(c_nx)}{(2c_n)^n},\quad c_n=\sqrt{\frac{n(n-1)}{2}}. \] They credit this result to \textit{G. Szegõ} [Orthogonal polynomials, Amer. Math. Soc. (1939; Zbl 0023.21505)], but their proof is different. They also give a recursive formula for the coefficients of \(P_n(x)\) and find the roots explicitly for \(n\le 7\). Third, the authors prove that \(\mathbf{ x}_n\) maximizes \(|v_n(\mathbf{x}_n)|\) over the real unit sphere if and only if \[ \sum_{\substack{i,j=1\\ i<j}}^n\frac{1}{(x_j-x_i)^2}=\frac{1}{2}\Big(\frac{n(n-1)}{2}\Big)^2,\quad \sum_{i=1}^nx_i^2=1. \] Fourth, they study limits involving these matrices or determinants. For example, they show that \[ G_{mn}(\mathbf{x}_n,\mathbf{a}_m)=\lim_{k\to\infty}V_{km}(\mathbf{a}_m)D_kV_{kn}(\log{\mathbf{x}_n}), \] where the limit is entrywise, \[ D_k=\textrm{diag}\,\Big(\frac{1}{0!},\frac{1}{1!},\dots,\frac{1}{(k-1)!}\Big), \] each \(x_j\ne 0\), and \(\log{\mathbf{x}_n}=(\log{x_1},\dots,\log{x_n})\). (In the complex case, the branch of logarithm is fixed and defines \(x_j^{a_i}=e^{a_i\log{x_j}}\).) For the entire collection see [Zbl 1467.16001].