Asymptotic Distribution of the Zeros of recursively defined\n Non-Orthogonal Polynomials

2021/07/11 by Bernhard Heim, Markus Neuhäuser, Heim, Bernhard +1
Mathematics · #11B37 #30C15 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2107.05013

openalex publication_date 2021/07/11 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We study the zero distribution of non-orthogonal polynomials attached to\ng(n)=s(n)=n²: \Q_n^g(x)= x
sum_k=1ⁿ g(k)
,\nQ_n-k^g(x),
quad Q₀^g(x):=1. It is known that the case\ng=id involves Chebyshev polynomials of the second kind. The zeros of\nQ_n^s(x) are real, simple, and are located in (-6\√(3),0]. Let\nN_n(a,b) be the number of zeros between -6 \√(3) \≤ a < b \≤ 0. Then\nwe determine a density function v(x), such that \
lim_{n\n
rightarrow
infty}
fracN_n(a,b)n =
int_a^b v(x)
,
,
mathrmdx.\n The polynomials Q_n^s(x) satisfy a four-term recursion. We\npresent in detail an analysis of the fundamental roots and give an answer to an\nopen question on recent work by Adams and Tran--Zumba. We extend a method\nproposed by Freud for orthogonal polynomials to more general systems of\npolynomials. We determine the underlying moments and density function for the\nzero distribution.\n

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