Log-Concavity in Powers of Infinite Series Close to $(1-z)^{-1}$

DOI10.1007/S40993-022-00370-5arXiv2203.12008WikidataQ114217975 ScholiaQ114217975MaRDI QIDQ6394425

Publication date: 22 March 2022

Abstract: In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If

f (z) = s u m_{n} a_{n} z^{n}

is an infinite series with

a_{n} g e q 1

and

a_{0} + c d o t s + a_{n} = O (n + 1)

for all

n

, we prove that a super-polynomially long initial segment of

f^{k} (z)

is log-concave. Furthermore, if there exists constants

C > 1

and

a l p h a < 1

such that

a_{0} + c d o t s + a_{n} = C (n + 1) - R_{n}

where

0 l e q R_{n} l e q O ((n + 1)^{a l p h a})

, we show that an exponentially long initial segment of

f^{k} (z)

is log-concave. This resolves a conjecture proposed by Letong Hong and the author, which implies another conjecture of Heim and Neuhauser that the Nekrasov-Okounkov polynomials

Q_{n} (z)

are unimodal for sufficiently large

n

.

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Special sequences and polynomials (11B83) Analytic theory of partitions (11P82)

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