Log-concavity in powers of infinite series close to $$(1 - z)^{-1}$$ ( 1 - z ) - 1

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.