Exceptional zeros of polynomials satisfying a three-term recurrence

Duc, Anh

doi:10.1016/j.jpaa.2017.04.017

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“…With a bit more work on the root distribution of the polynomial W 3 (z) in Theorem 4.3, one may obtain Theorem 5.1. We notice that the polynomial sequence {W n (z)} n satisfying Recurrence (3.1) can be written as W n (z) = f n (z) + g n (z), where {f n (z)} n is a polynomial sequence whose root distribution has been studied by Tran [25] and {g n (z)} n is a polynomials sequence whose root distribution has been studied by Mai [17]. We did not find a modification of any proof of theirs which may show Theorem 3.3.…”

Section: Discussionmentioning

confidence: 99%

Piecewise interlacing zeros of polynomials

Wang¹,

Zhang²

2017

Preprint

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We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly speaking, one constructs a collection of disjoint intervals such that one may show by induction that consecutive polynomials have interlacing zeros over each of the intervals. We confirm the real-rootedness of some polynomials satisfying a recurrence with linear polynomial coefficients. This extends Gross et al.'s work where one of the polynomial coefficients is a constant.

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Section: Discussionmentioning

confidence: 99%