The Rogers–Ramanujan–Gordon theorem for overpartitions

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–G...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2013-06, Vol.106 (6), p.1371-1393
Main Authors: Chen, William Y. C., Sang, Doris D. M., Shi, Diane Y. H.
Format: Article
Language:English
Subjects:
Analogue
Congruences
Functions (mathematics)
Mathematical analysis
Partitions
Theorems
Citations: Items that cite this one
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Summary: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.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms/pds056