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

For $\ell\geq 2$, 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 Tu\'{r}an inequalities for the $\ell$-regular overpartition function using a result of Griffin, Ono, Rolen, and Zagier.