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<p>The Rogers–Ramanujan identities, which were first introduced by Rogers and then rediscovered by Ramanujan, have attracted a lot of attention. In 1961, Gordon considered the combinatorial generalization of these two identities. Then, in view of the <i>q</i>-difference equations for certain basic hypergeometric series, Andrews derived an analytic version of Gordon’s theorem involving multiple series. Later, Bressoud established a companion for even moduli. Then, subsequent research has been focused on overpartition analogues. In this paper, with the aid of some <i>q</i>-difference equations, we establish the overpartition pair analogues of the aforementioned two theorems due to Gordon and Bressoud. Meanwhile, the corresponding multiple series identities of the Rogers–Ramanujan type are derived by the Bailey pair method. Furthermore, we use the Gordon markings of overpartition pairs to give the combinatorial interpretations of the multiple series.</p>

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Multiple Series Identities of the Rogers–Ramanujan Type for Overpartition Pairs

  • 期刊论文

  • 发表日期：

    2025年1月27日

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  • Nancy S. S. Gu,
  • Dan-Tong Wang

摘要

The Rogers–Ramanujan identities, which were first introduced by Rogers and then rediscovered by Ramanujan, have attracted a lot of attention. In 1961, Gordon considered the combinatorial generalization of these two identities. Then, in view of the q-difference equations for certain basic hypergeometric series, Andrews derived an analytic version of Gordon’s theorem involving multiple series. Later, Bressoud established a companion for even moduli. Then, subsequent research has been focused on overpartition analogues. In this paper, with the aid of some q-difference equations, we establish the overpartition pair analogues of the aforementioned two theorems due to Gordon and Bressoud. Meanwhile, the corresponding multiple series identities of the Rogers–Ramanujan type are derived by the Bailey pair method. Furthermore, we use the Gordon markings of overpartition pairs to give the combinatorial interpretations of the multiple series.