The Ramanujan Journal ( IF 0.7 ) Pub Date : 2020-05-15 , DOI: 10.1007/s11139-019-00228-y Nancy S. S. Gu , Chen-Yang Su
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.
中文翻译：
$$ M_2 $$ M2-模数为4和8的过度划分的等级
Overpartition是一个分区，在该分区中，数字的首次出现可能会被覆盖。对于分区\（\ lambda \），让\（\ ell（\ lambda）\）表示\（\ lambda \）的最大部分，让\（n（\ lambda）\）表示其部分数。然后，将分区的\（M_2 \） -rank定义为
$$ \ begin {aligned} M_2 \ text {-rank}（\ lambda）：= \ left \ lceil \ frac {\ ell（\ lambda）} {2} \ right \ rceil -n（\ lambda）+ n（ \ lambda _0）-\ chi（\ lambda），\ end {aligned} $$其中\（\志（\拉姆达）= 1 \）如果\（\ ELL（\拉姆达）\）是奇数和非上划线和\（\志（\拉姆达）= 0 \） ，否则。在本文中，我们研究了\（M_2 \） overpartitions的秩差异模4和8特别是，我们得到的生成函数之间的一些关系\（M_2 \）秩差异模4和8，而第二级模拟theta函数。此外，我们推导了\（M_2 \）-过度分区的秩的一些不等式。
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