Rademacher-type formula and higher order Turán inequalities for $\ell$-regular overpartitions

Abstract

For $\ell \ge 2$, let ${\overline{A}}_{\ell }(n)$ count the number of overpartitions of $n$ with no parts divisible by $\ell$. In this article, we employ the circle method to derive a Rademacher-type formula for ${\overline{A}}_{\ell }(n)$, when $\ell$ is a squarefree odd integer. As an application, we derive higher order Tuŕan inequalities for the $\ell$-regular overpartition function using a result of Griffin, Ono, Rolen, and Zagier.

23 pages, 1 figure