Ordinary, basic, and elliptic hypergeometric series - weighted enumeration and convolutions

We give a brief survey on the three-level hierarchy of hypergeometric series: ordinary (or rational) hypergeometric series, basic (or trigonometric, or q-) hypergeometric series, and, on the top level, elliptic hypergeometric series. The three different types of hypergeometric series satisfy a number of well known identities. Arguably the most fundamental identity for single series is the 10-V-9 summation formula (discovered by Frenkel and Turaev in 1997). This powerful elliptic hypergeometric series identity includes many of the other classical summations as special or limiting cases, including the well-known q-Chu-Vandermonde summation, an identity which can be easily interpreted combinatorially and corresponds to a convolution of q-binomial coefficients. We give a similar combinatorial interpretation for the 10-V-9 summation, now as a convolution of elliptic binomial coefficients. This is achieved by a weighted enumeration of lattice paths in a suitable lattice path model where we choose the weights to be specific elliptic functions. The weighted lattice path interpretation corresponds to a formulation entirely in algebraic terms; we describe an algebra of so called "elliptic commuting variables" in which the elliptic binomial coefficients appear as the normal form coefficients of a binomial. While our focus in this talk is on identities for basic and elliptic hypergeometric series (and part of our motivation for giving this talk is to make the area of elliptic hypergeometric series better known, in particular to researchers working in algebraic combinatorics), other suitable choices of the weights in our convolutions of weighted binomial coefficients yield convolutions of symmetric functions.