Let $B_{ k,i}(n)$ be the number of partitions of $n$ with certain difference condition and let $A_{ k,i}(n)$ be the number of partitions of $n$ with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that $B_{ k,i}(n)=A_{ k,i}(n)$. Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases $i=1$ and $i=k$. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let $D_{ k,i}(n)$ be the number of overpartitions of $n$ satisfying certain difference condition and $C_{ k,i}(n)$ be the number of overpartitions of $n$ whose non-overlined parts satisfy certain congruences condition. We show that $C_{ k,i}(n)=D_{ k,i}(n)$. By using a function introduced by Andrews, we obtain a recurrence relation which implies that the generating function of $D_{ k,i}(n)$ equals the generating function of $C_{ k,i}(n)$. We also find a generating function formula of $D_{ k,i}(n)$ by using Gordon marking representations of overpartitions, which can be considered as an overpartition analogue of an identity of Andrews for ordinary partitions.
Categories: math.CO math.NT