Log-concavity for partitions without sequences

TL;DR

This paper proves log-concavity and higher Turán inequalities for the partition function counting partitions without sequences, using advanced analytic and modular form techniques.

Contribution

It introduces a novel proof of log-concavity for this specific partition function and establishes higher Turán inequalities through new analytic estimates.

Findings

Proves log-concavity of the partition function without sequences.

Establishes higher Turán inequalities asymptotically.

Develops explicit estimates on modified Kloosterman sums.

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 Tur\'an 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.