The Rogers-Ramanujan-Gordon Theorem for Overpartitions
William Y.C. Chen, Doris D. M. Sang, and Diane Y. H. Shi
Abstract: Let Bk,i(n) be the number of partitions of n with certain difference condition and let Ak,i(n) be the number of partitions of n with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that Bk,i(n) = Ak,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 Dk,i(n) be the number of overpartitions of n satisfying certain difference condition and Ck,i(n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruences condition. We show that Ck,i(n) = Dk,i(n). By using a function introduced by Andrews, we obtain a recurrence relation which implies that the generating function of Dk,i(n) equals the generating function of Ck,i(n). We also find a generating function formula of Dk,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.
AMS Classification: 05A17, 11P84
Keywords: overpartition, the Rogers-Ramanujan-Gordon theorem, the Gordon marking of an overpartition