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PreprintCC BY 4.0arXiv

M⃗ Versions of Andrews-Gordon Identities Revisited

  • Alexander Berkovich
  • Aritram Dhar
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Abstract

In this paper, we revisit the work of Berkovich and Paule on variants of the Andrews-Gordon identities. We find a generalization of their principal polynomial identity and, as a consequence, obtain new M⃗ versions of the Andrews-Gordon identities. More precisely, for non-negative integers M1≥ M2≥ M3≥…≥ Mν, we show that a broad class of multi-sums of the form ∑n ( q N12+⋯+Nν2 − M1N1 − M2N2 − ⋯ − MνNν )/( (q)n1(q)n2⋯(q)nν ) can be expressed as a sum of products. Above, we use standard notations for q-Pochhammer symbols and Ni = ∑k=iνnk for 1≤ i≤ ν.

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Cite

BibTeXRIS
BibTeX
@misc{berkovich2026versions,
  title = {{$\vec M$ Versions of Andrews-Gordon Identities Revisited}},
  author = {Alexander Berkovich and Aritram Dhar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07490},
  eprint = {2609.07490},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07490},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - M⃗ Versions of Andrews-Gordon Identities Revisited
AU  - Alexander Berkovich
AU  - Aritram Dhar
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.07490
UR  - https://arxiv.org/abs/2609.07490
AB  - In this paper, we revisit the work of Berkovich and Paule on variants of the Andrews-Gordon identities. We find a generalization of their principal polynomial identity and, as a consequence, obtain new M⃗ versions of the Andrews-Gordon identities. More precisely, for non-negative integers M_1≥ M_2≥ M_3≥…≥ M_ν, we show that a broad class of multi-sums of the form ∑_n ( q^ N_1^2+⋯+N_ν^2 − M_1N_1 − M_2N_2 − ⋯ − M_νN_ν )/( (q)_n_1(q)_n_2⋯(q)_n_ν ) can be expressed as a sum of products. Above, we use standard notations for q-Pochhammer symbols and N_i = ∑_k=i^νn_k for 1≤ i≤ ν.
ER  -