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). By introducing the Gordon marking of an overpartition, we find a generating function formula for D_k,i(n) that can be considered 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

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