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 D 4 . Choosing a + , a − positive and \({\bf \gamma},\hat{v}\) real, we obtain detailed information on the |Re v|→∞ 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.