Linear relations for the number of overpartitions into odd parts

Let ${\bar{p}}_o(n)$ denote the number of overpartitions of $n$ into odd parts. The partition function ${\bar{p}}_o(n)$ has been the subject of many recent studies where many explicit Ramanujan-like congruences were discovered. In this paper, we provide three linear recurrence relation for ${\bar{p}}_o(n)$. Several connections with partitions into parts not congruent to $2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, overpartitions and partitions into distinct parts are presented in this context.