Bilateral Bailey Lattices and Andrews-Gordon Type Identities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918002669125632 |
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| author | Dousse, Jehanne Jouhet, Frédéric Konan, Isaac |
| author_facet | Dousse, Jehanne Jouhet, Frédéric Konan, Isaac |
| contents | We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative to $a/q$. Using this and similar lemmas, we give bilateral versions and simple proofs of other (new and known) Bailey lattices, including a Bailey lattice of Warnaar and the inverses of Bailey lattices of Lovejoy. As consequences of our bilateral point of view, we derive new $m$-versions of the Andrews-Gordon identities, Bressoud's identities, a new companion to Bressoud's identities, and the Bressoud-Göllnitz-Gordon identities. Finally, we give a new elementary proof of another very general identity of Bressoud using one of our Bailey lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02346 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bilateral Bailey Lattices and Andrews-Gordon Type Identities Dousse, Jehanne Jouhet, Frédéric Konan, Isaac Number Theory Combinatorics 11P84, 05A30, 33D15, 33D90 We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative to $a/q$. Using this and similar lemmas, we give bilateral versions and simple proofs of other (new and known) Bailey lattices, including a Bailey lattice of Warnaar and the inverses of Bailey lattices of Lovejoy. As consequences of our bilateral point of view, we derive new $m$-versions of the Andrews-Gordon identities, Bressoud's identities, a new companion to Bressoud's identities, and the Bressoud-Göllnitz-Gordon identities. Finally, we give a new elementary proof of another very general identity of Bressoud using one of our Bailey lattices. |
| title | Bilateral Bailey Lattices and Andrews-Gordon Type Identities |
| topic | Number Theory Combinatorics 11P84, 05A30, 33D15, 33D90 |
| url | https://arxiv.org/abs/2307.02346 |