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.

Bringmann，Lovejoy和Osburn给出了分区的等级和曲柄的定义。让\（\ overline {N}（s，l; n）\）（resp。\（\ overline {M}（s，l; n）\），\（\ overline {M2}（s，l; n ）\））表示n的超分区数，其秩（分别为第一个残差曲柄，第二个残差曲柄）与s模l一致。确定以3、5、6、7和10为模的超分区的秩差。在本文中，我们为\（\ overline {N}（s，l; n）\），\（\ overline {M}（s，l; n）\）和\（\ overline {M2 }（s，l; n）\）与\（l = 4，8 \）通过利用Appell-Lerch的总和和theta函数恒等式。此外，根据这些生成函数，我们得到模数为4和8的过分配的秩和曲柄上的一些等式和不等式。