Asymptotic distribution of the zeros of recursively defined non-orthogonal polynomials

Let g be a normalized arithmetic function. We define polynomials Q(n)(g)(x)=x Sigma(n)(k=1) g(k) Q(n-k)(g)(x), Q(0)(g)(x) := 1. It is known that the case g = id involves Chebyshev polynomials of the second kind Q(n)(id)(x) = x Un-1(x/2 +1). In this paper we study the zero distribution of the non-orthogonal polynomials associated ./ with s (n) =n(2). We show that the zeros of Q(n)(s)(x) are real, simple, and are located in (-6 root 3, 0]. Let N-n(a,b) be the number of zeros between -6 root 3 <= a <= b <= 0. Then we determine a density function v(x), such that lim N-n ->infinity(n)(a, b)/n integral(b)(a) = v(x) dx. 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. (c) 2022 Elsevier Inc. All rights reserved.