Proof of a summation formula for an ${\tilde{A}}_n$ basic hypergeometric series conjectured by Warnaar

Abstract

A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system ${\tilde{A}}_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's ${}_1{\psi }_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems ${\tilde{A}}_n$ respectively ${\tilde{A}}_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.

9 pages, AmS-LaTeX