$$M_2$$-Ranks of overpartitions modulo 4 and 8

An overpartition is a partition in which the first occurrence of a number may be overlined. For an overpartition $$\lambda $$ , let $$\ell (\lambda )$$ denote the largest part of $$\lambda $$ , and let $$n(\lambda )$$ denote its number of parts. Then the $$M_2$$ -rank of an overpartition is defined as $$\begin{aligned} M_2\text {-rank}(\lambda ):=\left\lceil \frac{\ell (\lambda )}{2}\right\rceil -n(\lambda )+n(\lambda _0)-\chi (\lambda ), \end{aligned}$$ where $$\chi (\lambda )=1$$ if $$\ell (\lambda )$$ is odd and non-overlined and $$\chi (\lambda )=0$$ , otherwise. In this paper, we study the $$M_2$$ -rank differences of overpartitions modulo 4 and 8. Especially, we obtain some relations between the generating functions of the $$M_2$$ -rank differences modulo 4 and 8 and the second order mock theta functions. Furthermore, we deduce some inequalities on $$M_2$$ -ranks of overpartitions.