An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Schlosser, Michael J.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/213524
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem. Michael J. Schlosser. SIGMA 21 (2025), 052, 15 pages

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.