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

Abstract:

Given ${x_1} < {x_2} < \cdots < {x_n}$ and ${y_1} < {y_2} < \cdots < {y_{n - 1}}$, two interlacing sequences of real numbers, the rectangular diagram for these numbers is a continuous piecewise linear function with slopes $\pm 1$ and with n local minima at the points ${x_i}$ and $n - 1$ local maxima at the points ${y_j}$. 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 $(C, - 1)$-summability results on distribution of zeros of orthogonal polynomials proved by us some time ago.