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


Authors: Paul Nevai and Walter Van Assche
Journal: Proc. Amer. Math. Soc. 122 (1994), 759-767
MSC: Primary 42C05
MathSciNet review: 1204382
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1204382-X
Keywords: Orthogonal polynomials, zeros, rectangular diagrams
Article copyright: © Copyright 1994 American Mathematical Society