A Generalized Hypergeometric Function II. Asymptotics and D4 Symmetry

In previous work we introduced and studied a function $R(a_{+},a_{-},{\bf c};v,\hat{v})$ that generalizes the hypergeometric function. In this paper we focus on a similarity-transformed function ${\mathcal E} (a_{+},a_{-},\gamma ;v,\hat{v})$, with parameters γ∈ℂ⁴ related to the couplings c∈ℂ⁴ by a shift depending on a₊, a_-. We show that the ℰ-function is invariant under all maps γ↦w(γ), with w in the Weyl group of type D₄. Choosing a₊, a_- positive and ${\bf \gamma},\hat{v}$ real, we obtain detailed information on the |Rev|→∞ asymptotics of the ℰ-function. In particular, we explicitly determine the leading asymptotics in terms of plane waves and the c-function that implements the similarity R→ℰ.