Some Congruences for Overpartitions with Restriction

For any given positive integers $m$ and $n$, let $\overline{p}_m(n)$ denote the number of overpartitions of $n$ with no parts divisible by $4m$ and only the parts congruent to $m$ modulo $2m$ overlined. In this paper, we prove Ramanujan-type congruences modulo 2 for $\overline{p}_m(n)$ by applying $q$-series and Ramanujan's theta-function identities.