An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem
We introduce an algebra of elliptic commuting variables involving a base , nome , and 2 noncommuting variables. This algebra, which for = 1 reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of -commuting variables. We present a multinomial t...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2025 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/213524 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem. Michael J. Schlosser. SIGMA 21 (2025), 052, 15 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862573418705059840 |
|---|---|
| author | Schlosser, Michael J. |
| author_facet | Schlosser, Michael J. |
| citation_txt | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem. Michael J. Schlosser. SIGMA 21 (2025), 052, 15 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce an algebra of elliptic commuting variables involving a base , nome , and 2 noncommuting variables. This algebra, which for = 1 reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of -commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstraß type elliptic partial fraction decomposition. From the elliptic multinomial theorem, we obtain, by convolution, an identity equivalent to Rosengren's type extension of the Frenkel-Turaev ₁₀₉ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice ℤʳ, this derivation of Rosengren's ᵣ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.
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| first_indexed | 2026-03-21T12:06:12Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-213524 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:06:12Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Schlosser, Michael J. 2026-02-18T11:24:52Z 2025 An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem. Michael J. Schlosser. SIGMA 21 (2025), 052, 15 pages 1815-0659 2020 Mathematics Subject Classification: 05A10; 11B65; 33D67; 33D80; 33E90 arXiv:2307.12921 https://nasplib.isofts.kiev.ua/handle/123456789/213524 https://doi.org/10.3842/SIGMA.2025.052 We introduce an algebra of elliptic commuting variables involving a base , nome , and 2 noncommuting variables. This algebra, which for = 1 reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of -commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstraß type elliptic partial fraction decomposition. From the elliptic multinomial theorem, we obtain, by convolution, an identity equivalent to Rosengren's type extension of the Frenkel-Turaev ₁₀₉ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice ℤʳ, this derivation of Rosengren's ᵣ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity. The author’s research was partly supported by an FWF Austrian Science Fund grant, doi:10.55776/ P32305. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem Article published earlier |
| spellingShingle | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem Schlosser, Michael J. |
| title | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem |
| title_full | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem |
| title_fullStr | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem |
| title_full_unstemmed | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem |
| title_short | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem |
| title_sort | algebra of elliptic commuting variables and an elliptic extension of the multinomial theorem |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213524 |
| work_keys_str_mv | AT schlossermichaelj analgebraofellipticcommutingvariablesandanellipticextensionofthemultinomialtheorem AT schlossermichaelj algebraofellipticcommutingvariablesandanellipticextensionofthemultinomialtheorem |