Asymptotic Normality of the Coefficients of the Morgan-Voyce Polynomials

TL;DR

This paper investigates the asymptotic behavior of Morgan-Voyce polynomial coefficients, establishing their normal distribution tendencies through advanced probabilistic methods and linking them to Fibonacci numbers.

Contribution

It introduces a new analysis of Morgan-Voyce polynomial coefficients, proving their asymptotic normality and deriving related limit theorems using sophisticated probabilistic techniques.

Findings

Coefficients follow a central limit theorem.

Coefficients satisfy a local limit theorem.

Results involve Fibonacci number connections.

Abstract

We study arithmetic and asymptotic properties of polynomials provided by $Q_n(x):= x \sum_{k=1}^n k \, Q_{n-k}(x)$ with initial value $Q_0(x)=1$. The coefficients satisfy a central limit theorem and a local limit theorem involving Fibonacci numbers. We apply methods of Berry and Esseen, Harper, Bender, and Canfield.