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

Abstract

AbstractIn 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.