2023/07/24 by Michael J. Schlosser, Schlosser, Michael J.
Mathematics · #33D67 #33D80 #33E90 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Primary 05A10 #Quantum Algebra (math.QA) #Secondary 11B65
paper · pdf · doi:10.48550/arxiv.2307.12921
openalex publication_date 2023/07/24 · openalex created_date 2023/07/26 · openalex updated_date 2026/08/01
We introduce an algebra of elliptic commuting variables involving a base q, nome p, and 2r noncommuting variables. This algebra, which for r=1 reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of r q-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 \mathsf A elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type \mathsf A extension of the Frenkel-Turaev 10V9 summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice \mathbb Zr, this derivation of Rosengren's \mathsf Ar Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.