Log-concavity for partitions without sequences

TL;DR

The paper proves that $p_2(n)$, the number of partitions of $n$ with no consecutive parts, is log-concave for all sufficiently large $n$ (with a finite set of small cases checked directly) by deriving a strong asymptotic expansion of $p_2(n)$ from a mixed-mock modular form formula and explicit Kloosterman-sum bounds. It provides a 9-term explicit asymptotic expansion with coefficients $a_k$, obtained via a saddle-point analysis of a Bessel-type integral, and uses this to verify log-concavity through careful error control. Building on this, the authors apply the Griffin–Ono–Rolen–Zagier criterion to establish higher Turán inequalities, showing that for any fixed $d\ge3$ the Jensen polynomials $J_{p_2}^{d,n}$ become hyperbolic for all sufficiently large $n$, i.e., the corresponding Turán inequalities hold asymptotically. These results extend the landscape of log-concavity and Turán-type properties to partitions without sequences and demonstrate the effectiveness of exact mixed-mock modular formulas coupled with analytic machinery in detecting convexity phenomena in partition functions.

Abstract

We prove log-concavity for the function counting partitions without sequences. We use an exact formula for a mixed-mock modular form of weight zero, explicit estimates on modified Kloosterman sums and analytic techniques. Finally, we establish the higher Turán inequalities in an asymptotic form of the aforementioned partition function using a well established criterion of Griffin, Ono, Rolen, and Zagier on the zeros of Jensen polynomials.

Theorems & Definitions (37)

• Theorem
• Theorem
• Lemma 2.1
• Theorem 2.2: Weil bound for Kloosterman sums
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof