An elliptic extension of the multinomial theorem

Abstract: We present a multinomial theorem for elliptic commuting variables. This result extends the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a simple elliptic star-triangle relation, ensuring the uniqueness of the normal form coefficients, and, for the recursion of the closed form elliptic multinomial coefficients, the Weierstra{\ss} type $\mathsf A$ elliptic partial fraction decomposition. From our elliptic multinomial theorem we obtain, by convolution, an identity that is equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel--Turaev ${}_{10}V_9$ summation, which in the trigonometric or basic limiting case reduces to Milne's type $\mathsf A$ extension of the Jackson ${}_8\phi_7$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, our derivation of the $\mathsf A_r$ Frenkel--Turaev summation constitutes the first combinatorial proof of that fundamental identity, and, at the same time, of important special cases including the $\mathsf A_r$ Jackson summation.