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Majorization results for zeros of orthogonal polynomials

Author: Walter Van Assche
Journal: Proc. Amer. Math. Soc. 145 (2017), 3849-3863
MSC (2010): Primary 33C45, 42C05; Secondary 26C10, 30C15, 15B51
Published electronically: May 24, 2017
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Abstract: We show that the zeros of consecutive orthogonal polynomials $ p_n$ and $ p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of $ p_n$ and the associated polynomial $ p_{n-1}^{(1)}$ and for the zeros of the polynomial obtained by deleting the $ k$th row and column $ (1 \leq k \leq n)$ in the corresponding Jacobi matrix.

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Additional Information

Walter Van Assche
Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Keywords: Orthogonal polynomials, zeros, doubly stochastic matrices
Received by editor(s): July 12, 2016
Published electronically: May 24, 2017
Additional Notes: This research was supported by KU Leuven research grant OT/12/073 and FWO research project G.0864.16N
Dedicated: Dedicated to T.J. Stieltjes on the occasion of his 160th birthday
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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