We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the Macdonald identities for the seven infinite families of irreducible reduced affine root systems.

We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall-Littlewood symmetric functions.

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 φ 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.

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We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For motivation, we review our previous simple proof (Proc. Amer. Math. Soc. 130 (2002), 1103-1111) of Bailey's very-wellpoised 6 ψ 6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8 ψ 8 transformation, a very-well-poised 10 ψ 10 transformation, by induction, Slater's general transformation for very-well-poised 2r ψ 2r series, and Slater's transformation for general r ψr series. Finally, we derive some new transformations for bilateral basic hypergeometric series of a specific type. Mathematics Subject Classification (2000). Primary 33D15.