Log-concavity in powers of infinite series close to $$(1 - z)^{-1}$$ ( 1 - z ) - 1
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40993_2022_Article_370.pdf
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299.13 KB
Format
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Author(s)
Zhang, Shengtong
Date Issued
September 12, 2022
Publisher
Springer International Publishing
Citation
Research in Number Theory. 2022 Sep 12;8(4):66
Version
Final published version
Abstract
Abstract
In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If
$$f(z) = \sum _n a_nz^n$$
f
(
z
)
=
∑
n
a
n
z
n
is an infinite series with
$$a_n \ge 1$$
a
n
≥
1
and
$$a_0 + \cdots + a_n = O(n + 1)$$
a
0
+
⋯
+
a
n
=
O
(
n
+
1
)
for all n, we prove that a super-polynomially long initial segment of
$$f^k(z)$$
f
k
(
z
)
is log-concave. Furthermore, if there exists constants
$$C > 1$$
C
>
1
and
$$\alpha < 1$$
α
<
1
such that
$$a_0 + \cdots + a_n = C(n + 1) - R_n$$
a
0
+
⋯
+
a
n
=
C
(
n
+
1
)
-
R
n
where
$$0 \le R_n \le O((n + 1)^{\alpha })$$
0
≤
R
n
≤
O
(
(
n
+
1
)
α
)
, we show that an exponentially long initial segment of
$$f^k(z)$$
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
$$Q_n(z)$$
Q
n
(
z
)
are unimodal for sufficiently large n.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s40993-022-00370-5
