# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocalHTML articles powered by AMS MathViewer

by Tamás Erdélyi
Trans. Amer. Math. Soc. 374 (2021), 3077-3091 Request permission

## Abstract:

Let \begin{equation*} {\mathcal {K}}_n \coloneq \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \quad a_k \in {\mathbb {C}} , \quad |a_k| = 1 \right \} . \end{equation*} 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 \begin{equation*} \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} \end{equation*} 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 \begin{equation*} \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} \end{equation*} 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.
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