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Keywords:

orthogonal polynomials, zeros, rectangular diagrams, Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, ORTHOGONAL POLYNOMIALS, ZEROS, RECTANGULAR DIAGRAMS, 0101 Pure Mathematics, 4904 Pure mathematics

Abstract:

Given x(1) < / x(2) < /...< / x(n) and y(1) < / y(2) < /...< / y(n-1), two interlacing sequences of real numbers, the rectangular diagram for these numbers is a continuous piecewise linear function with slopes +/-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.