Jensen polynomials associated with Wright's circle method: Hyperbolicity and Tur\'an inequalities

TL;DR

This paper demonstrates that Fourier coefficients from functions analyzed via Wright's circle method exhibit asymptotic hyperbolicity in their Jensen polynomials, leading to the satisfaction of all higher-order Turán inequalities, with applications in partition theory and number theory.

Contribution

It establishes the asymptotic hyperbolicity of Jensen polynomials for Fourier coefficients under Wright's circle method and proves they satisfy higher-order Turán inequalities, extending previous frameworks.

Findings

Jensen polynomials are asymptotically hyperbolic.

Fourier coefficients satisfy all higher-order Turán inequalities.

Applications to partition functions and number-theoretic functions.

Abstract

We study the Fourier coefficients of functions satisfying a certain version of Wright's circle method with finitely many major arcs. We show that the Jensen polynomials associated with such Fourier coefficients are asymptotically hyperbolic, building on the framework of Griffin--Ono--Rolen--Zagier and others. Consequently, we prove that the Fourier coefficients asymptotically satisfy all higher-order Tur\'an inequalities. As an application, we apply our results to both $(q^t;q^t)_\infty^{-r}$, which counts $r$-coloured partitions into parts divisible by $t$, and to the function $(q^{a};q^{p})_\infty^{-1}$ where $p$ is prime and $0\leq a<p$, a ubiquitous function throughout number theory.