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

TL;DR

This paper analyzes the asymptotic zero distribution of a class of recursively defined non-orthogonal polynomials related to quadratic functions, extending methods from orthogonal polynomial theory.

Contribution

It introduces a new approach to determine the zero distribution of non-orthogonal polynomials, extending Freud's method to more general polynomial systems.

Findings

Zeros are real, simple, and located in a specific interval.

A density function for zero distribution is explicitly determined.

The work answers an open question by Adams and Tran--Zumba.

Abstract

We study the zero distribution of non-orthogonal polynomials attached to $g(n)=s(n)=n^2$: \begin{equation*} Q_n^g(x)= x \sum_{k=1}^n g(k) \, Q_{n-k}^g(x), \quad Q_0^g(x):=1. \end{equation*} It is known that the case $g=id$ involves Chebyshev polynomials of the second kind. The zeros of $Q_n^s(x)$ are real, simple, and are located in $(-6\sqrt{3},0]$. Let $N_n(a,b)$ be the number of zeros between $-6 \sqrt{3} \leq a < b \leq 0$. Then we determine a density function $v(x)$, such that \begin{equation*} \lim_{n \rightarrow \infty} \frac{N_n(a,b)}{n} = \int_a^b v(x) \,\, \mathrm{d}x. \end{equation*} The polynomials $Q_n^s(x)$ satisfy a four-term recursion. We present in detail an analysis of the fundamental roots and give an answer to an open question on recent work by Adams and Tran--Zumba. We extend a method proposed by Freud for orthogonal polynomials to more general systems of polynomials. We determine the underlying moments and density function for the zero distribution.