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

Abstract

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 𝗔 elliptic partial fraction decomposition. From the elliptic multinomial theorem, we obtain, by convolution, an identity equivalent to Rosengren's type 𝗔 extension of the Frenkel-Turaev ₁₀𝑉₉ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice ℤʳ, this derivation of Rosengren's 𝗔ᵣ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.