The Rogers–Ramanujan–Gordon theorem for overpartitions

By William Y. C. Chen, Doris D. M. Sang, Diane Y. H. Shi. 2011

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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 congruence condition. We show that Ck, i(n)=Dk, i(n). By using a function introduced by Andrews, we obtain a recurrence relation that implies that the generating function of Dk, i(n) equals the generating function of Ck, i(n). By introducing the Gordon marking of an overpartition, we find a generating function formula for Dk, i(n) that can be considered an overpartition analogue of an identity of Andrews for ordinary partitions.

Subfields: Algebra and Number Theory, Applied Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/94f12eb897998b4e19002b90cfbfd63962d3d870