Remarks on the (𝐶,-1)-summability of the distribution of zeros of orthogonal polynomials

Abstract

Given x 1 > x 2 > ⋯ > x n {x_1} > {x_2} > \cdots > {x_n} and y 1 > y 2 > ⋯ > y n − 1 {y_1} > {y_2} > \cdots > {y_{n - 1}} , two interlacing sequences of real numbers, the rectangular diagram for these numbers is a continuous piecewise linear function with slopes ± 1 \pm 1 and with n local minima at the points x i {x_i} and n − 1 n - 1 local maxima at the points y j {y_j} . Recently, S. Kerov determined the asymptotic behavior of the rectangular diagrams associated with the zeros of two consecutive orthogonal polynomials for which the coefficients in the three-term recurrence relation converge. The purpose of this note is to show how this result of S. Kerov and even some of its generalizations follow directly from certain ( C , − 1 ) (C, - 1) -summability results on distribution of zeros of orthogonal polynomials proved by us some time ago.