2018/10/09 by Tamás Erdélyi, Erdélyi, Tamás · 1 citation
Mathematics · #11C08 #26C10 #30C15 #41A17 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1810.04287
openalex publication_date 2018/10/09 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
Let
mathcal Kn :=
left
pn: pn(z) =
sumk=0nak zk,
enspace\nak
in
mathbb C
,,
enspace |ak| = 1
right
,. A sequence (Pn) of\npolynomials Pn \∈ mathcal Kn is called ultraflat if (n +\n1)-1/2|Pn(eit)| converge to 1 uniformly in t \∈ mathbb R. In\nthis paper we prove that
frac12
pi
int02
pi
left| (Pn -\nPn^*)(eit)
right|q
, dt
sim
frac2q
Gamma
left(
fracq+12\n
right)
Gamma
left(
frac q2 + 1
right)
sqrt
pi
,
, nq/2 for every\nultraflat sequence (Pn) of polynomials Pn \∈ mathcal Kn and for\nevery q \∈ (0,\∞), where Pn^* is the conjugate reciprocal polynomial\nassociated with Pn, \Γ is the usual gamma function, and the \∼\nsymbol means that the ratio of the left and right hand sides converges to 1\nas n \→ \∞. Another highlight of the paper states that\n
frac12
pi
int02
pi
left| (Pn^
prime - Pn*
prime)(eit)\n
right|2
, dt
sim
frac2n33 for every ultraflat sequence (Pn) of\npolynomials Pn \∈ mathcal Kn. We prove a few other new results and\nreprove some interesting old results as well.\n