Elliptic enumeration of nonintersecting lattice paths

Abstract

We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's ${}_{10}{V}_9$ summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's ${}_8{\varphi }_7$ and Dougall's ${}_7{F}_6$ summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the ${}_{10}{V}_9$ summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives.