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

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