The Rogers-Ramanujan-Gordon Theorem for Overpartitions
William Y.C. Chen, Doris D. M. Sang, and Diane Y. H. Shi
Abstract: 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. AMS Classification: 05A17, 11P84 Keywords: overpartition, the Rogers-Ramanujan-Gordon theorem, the Gordon marking of an overpartition Download: PDF |