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

Let K n class="MJX-TeXAtom-REL"> ? { Q : ( z stretchy="false">) = ? k 0 a , ? ?<!-- ? </mml:msubsup> | ?<!-- ? <mml:mi>q d ?<!-- ? mathvariant="normal">?<!-- ? <mml:mo>( ) \frac {1}{2\pi } \int _0^{2\pi }{\left | (P_n - P_n^*)(e^{it}) |^q \, dt} \sim {{2}^q \Gamma (\frac {q+1}{2} )}{\Gamma q2 ) \sqrt {\pi }} \,\, n^{q/2} for every and alttext="q infinity mathvariant="normal">?<!-- ? encoding="application/x-tex">q (0,\infty ) , where asterisk"> encoding="application/x-tex">P_n^* the conjugate reciprocal polynomial associated with encoding="application/x-tex">P_n alttext="normal Gamma"> encoding="application/x-tex">\Gamma usual gamma function, alttext="tilde"> encoding="application/x-tex">\sim symbol means ratio left right hand sides converges as alttext="n right-arrow infinity"> stretchy="false">?<!-- ? encoding="application/x-tex">n \rightarrow \infty Another highlight states prime squared cubed 3 EndFraction"> class="MJX-variant" mathvariant="normal">?<!-- ? <mml:mn>3 }\int (P_n^\prime P_n^{*\prime })(e^{it}) |^2 {2n^3}{3} We few other new results reprove some interesting old well.