A Paley–Wiener theorem for the Θ-hypergeometric transform: the even multiplicity case

AbstractThe Θ-hypergeometric functions generalize the spherical functions on Riemannian symmetric spaces and the spherical functions on non-compactly causal symmetric spaces. In this paper we consider the case of even multiplicity functions. We construct a differential shift operator Dm with smooth coefficients which generates the Θ-hypergeometric functions from finite sums of exponential functions. We then use this fact to prove a Paley–Wiener theorem for the Θ-hypergeometric transform