Andrews' singular overpartitions with odd parts

Abstract

Recently singular overpartitions was defined and studied by G. E. Andrews. He showed that such partitions can be enumerated by $\overline{C}_{\delta,i}(n)$, the number of overpartitions of $n$ such that no part is divisible by $\delta$ and only parts $\equiv\pm i(mod \delta)$ may be overlined. In this paper, we establish several infinite families of congruences $\overline{CO}_{\delta,i}(n)$, the number of singular overpartitions of $n$ into odd parts such that no part is divisible by $\delta$ and only parts $\equiv\pm i\AFODMod{\delta}$ may be overlined. For example, for all $n\geq 0$ and $\alpha\geq0$, $\overline{CO}_{3,1}(4\cdot3^{\alpha+3}n+7\cdot3^{\alpha+2})\equiv 0 \pmod{8}$.