$M_2$-Ranks of overpartitions modulo $6$ and $10$

Abstract

In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let ${\overline{N}}_2(s,m,n)$ to be the number of overpartitions of $n$ whose $M_2$-rank is congruent to $s$ modulo $m$. For $M_2$-ranks modulo $3$, Lovejoy and Osburn derived the generating function of ${\overline{N}}_2(s,3,n)-{\overline{N}}_2(t,3,n)$, which implies the inequalities ${\overline{N}}_2(0,3,n)\ge {\overline{N}}_2(1,3,n)$. For $\ell =6,10$, we consider the generating function ${\overline{R}}_{s,t}(d,\ell )$ of the $M_2$-rank differences ${\overline{N}}_2(s,\ell ,\ell n/2+d)+{\overline{N}}_2(s+1,\ell ,\ell n/2+d)-{\overline{N}}_2(t,\ell ,\ell n/2+d)-{\overline{N}}_2(t+1,\ell ,\ell n/2+d)$. By the method of Lovejoy and Osburn, we derive a formula for ${\overline{R}}_{0,2}(d,6)$. This leads to the inequalities for $n\ge 0$, ${\overline{N}}_2(0,6,3n)\ge {\overline{N}}_2(2,6,3n)$ and ${\overline{N}}_2(0,6,3n+1)\ge {\overline{N}}_2(2,6,3n+1)$. Based on the valence formula for modular functions, we compute ${\overline{R}}_{0,4}(d,10)$ and ${\overline{R}}_{1,3}(d,10)$. In particular, we notice that the generating function ${\overline{R}}_{0,2}(2,6)$ can be expressed in terms of the third order mock theta function $\rho (q)$, and the generating functions ${\overline{R}}_{0,4}(4,10)$, ${\overline{R}}_{1,3}(1,10)$ and ${\overline{R}}_{1,3}(4,10)$ can also be expressed in terms of the tenth order mock theta functions $\varphi (q)$ and $\psi (q)$.