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

Let

${\mathcal {K}}_n \colonequals \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \enskip a_k \in {\mathbb{C}}\,, \enskip \vert a_k\vert = 1 \right \}\,.$

A sequence $(P_n)$ of polynomials $P_n \!\in \! {\mathcal {K}}_n$ is called ultraflat if $(n + 1)^{-1/2}\vert P_n(e^{it})\vert$ converge to $1$ uniformly in $t \!\in \! {\mathbb{R}}$. In this paper we prove that

$\frac {1}{2\pi } \int _0^{2\pi }{\left \vert (P_n - P_n^*)(e^{it}... ...+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2}$

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

$\frac {1}{2\pi }\int _0^{2\pi }{\left \vert (P_n^\prime - P_n^{*\prime })(e^{it}) \right \vert^2 \, dt} \sim \frac {2n^3}{3}$

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.