Jensen Polynomials Associated with Wright's Circle Method: Hyperbolicity and Turán Inequalities

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, and Zagier. Consequently, we prove that the Fourier coefficients asymptotically satisfy all higher-order Turán inequalities. As an application, we apply our results to both (q^t; q^t)^−r ∞, which counts r-colored partitions into parts divisible by t, and to the function (q^a; q^p)⁻¹ ∞ where p is prime and 0 ≤ a < p, a ubiquitous function throughout number theory.