Asymptotic normality of coefficients of P-recursive polynomial sequences
Zhongjie Li
TL;DR
This paper addresses the problem of establishing asymptotic normality for the coefficients of general P-recursive polynomial sequences. It develops a systematic approach that derives asymptotics from $f_n(1)$, $f_n'(1)$, and $f_n''(1)$ within a real-rooted, nonnegative coefficient framework and applies a Bender-type local/central limit theorem to conclude normality. The main contribution is a sufficient condition (Theorem t2.3) that yields explicit mean and variance growth and guarantees asymptotic normality when $\sigma_n^2\to\infty$, illustrated by two examples. The Apéry and Franel polynomials demonstrate the method, with explicit limits such as $\mu_n \sim \frac{-1+\sqrt{5}}{2}n$, $\sigma_n^2 \sim \frac{5-2\sqrt{5}}{5}n$ for Apéry, and $\mu_n\sim n/2$, $\sigma_n^2\sim n/12$ for Franel, underscoring broad applicability to combinatorial sequences.
Abstract
In recent years, the asymptotic normality of some famous combinatorial sequences has been the subject of extensive study. However, the methods used to prove the asymptotic normality of various combinatorial sequences differ significantly. In this paper, we present a sufficient condition for establishing the asymptotic normality of the coefficients of a general P-recursive polynomial sequence. Additionally, we provide two examples that illustrate the application of this sufficient condition.