Menke points on the real line and their connection to classical orthogonal polynomials (Q1044632)

Let \(q\) and \(p\) be two positive numbers. Analogously to the famous Stieltjes problem from electrostatics the authors study the extremal problem for the function \[ T_n(x_1,\dots,x_n,y_1,\dots,y_{n-1})= \prod_{i=1}^n (1-x_i)^p \prod_{j=1}^n\prod_{k=1}^{n-1}|x_j-y_k| \prod_{l=1}^{n-1}(1+y_l)^q \] over all configurations satisfying \[ -1=y_0<x_1<y_1<\dots<y_{n-1}<x_{n}<y_n=1. \] For the case of an even number of points the expression and the interlacing property are similar. A system \(X=\{x_j\}\), \(Y=\{y_k\}\) for which the maximum is attained is called a \((p,q)\)-Menke system for \([-1,1]\). It turns out that the Menke points are related to the zeros and extrema of Jacobi polynomials. Here is a typical result in that direction. Theorem. The points \(X\) of the Menke system are the zeros and \(Y\) are the extrema of the Jacobi polynomials \(P_n^{(\alpha,\beta)}\) on \([-1,1]\), where \(\alpha=p-1\), \(\beta=q-1\). Similar results are established for the half line and the whole real line.