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

Abstract

Let $${\mathcal K}_n := \left\{p_n: p_n(z) = \sum_{k=0}^n{a_k z^k}, \enspace a_k \in {\mathbb C}\,,\enspace |a_k| = 1 \right\}\,.$$ 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 $$\frac{1}{2\pi} \int_0^{2\pi}{\left| (P_n - P_n^*)(e^{it}) \right|^q \, dt} \sim \frac{{2}^q \Gamma \left(\frac{q+1}{2} \right)}{\Gamma \left(\frac q2 + 1 \right) \sqrt{\pi}} \,\, n^{q/2}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$ and for every $q \in (0,\infty)$, where $P_n^*$ 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 $n \rightarrow \infty$. Another highlight of the paper states that $$\frac{1}{2\pi}\int_0^{2\pi}{\left| (P_n^\prime - P_n^{*\prime})(e^{it}) \right|^2 \, dt} \sim \frac{2n^3}{3}$$ 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.