The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal

Abstract

Let $${\mathcal{K}}_n:=\left\{p_n:p_n(z)=\sum _{k=0}^n{a}_k{z}^k,a_k\in \mathbb{C},|a_k|=1\right\}.$$ A sequence $(P_n)$ of polynomials $P_n\in {\mathcal{K}}_n$ is called ultraflat if $(n+1{)}^{-1/2}|P_n(e^{it})|$ converge to $1$ uniformly in $t\in \mathbb{R}$. In this paper we prove that $$\frac{1}{2\pi }{\int }_0^{2\pi }{\left|(P_n-P_n^{\ast })(e^{it})\right|}^{q}dt\sim \frac{2^q\Gamma \left(\frac{q+1}{2}\right)}{\Gamma \left(\frac{q}{2}+1\right)\sqrt{\pi }}{n}^{q/2}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n\in {\mathcal{K}}_n$ and for every $q\in (0,\infty )$, where $P_n^{\ast }$ is the conjugate reciprocal polynomial associated with $P_n$, $\Gamma$ is the usual gamma function, and the $\sim$ symbol means that the ratio of the left and right hand sides converges to $1$ as $n\to \infty$. Another highlight of the paper states that $$\frac{1}{2\pi }{\int }_0^{2\pi }{\left|(P_n^{\prime }-P_n^{\ast \prime })(e^{it})\right|}^{2}dt\sim \frac{2n^3}{3}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n\in {\mathcal{K}}_n$. We prove a few other new results and reprove some interesting old results as well.