Ranks, cranks for overpartitions and Appell–Lerch sums

The definitions of the rank and crank for overpartitions were given by Bringmann, Lovejoy and Osburn. Let $$\overline{N}(s,l;n)$$ (resp. $$\overline{M}(s,l;n)$$ , $$\overline{M2}(s,l;n)$$ ) denote the number of overpartitions of n with rank (resp. the first residual crank, the second residual crank) congruent to s modulo l. The rank differences of overpartitions modulo 3, 5, 6, 7 and 10 were determined. In this paper, we establish the generating functions for $$\overline{N}(s,l;n)$$ , $$\overline{M}(s,l;n)$$ and $$\overline{M2}(s,l;n)$$ with $$l=4, 8$$ by utilizing Appell–Lerch sums and theta function identities. Moreover, in light of these generating functions, we obtain some equalities and inequalities on ranks and cranks of overpartitions modulo 4 and 8.