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

TL;DR

This paper develops an algebraic framework of elliptic commuting variables, extending the multinomial theorem to higher rank and providing a combinatorial proof of a key summation identity in elliptic hypergeometric series.

Contribution

It introduces a new elliptic algebra of higher rank, extending previous binomial theorems, and connects algebraic identities with combinatorial lattice path interpretations.

Findings

Derived a higher-rank elliptic multinomial theorem.

Established a combinatorial proof of Rosengren's elliptic summation.

Linked algebraic identities to lattice path counting.

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 $\mathsf A$ 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 ${}_{10}V_9$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, this derivation of Rosengren's $\mathsf A_r$ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.