The Askey-Wilson Function is the Laplace Transform of a Positive Measure

The Askey-Wilson function $ϕ(x, γ)$ is a meromorphic continuation of the Askey-Wilson polynomials in their spectral variable off the lattice $γ= \tilde{a} q^m, m \in \mathbb{Z}_+$. We show that the analytic part of the Askey-Wilson function $ϕ^{an}(x, \tilde{a} q^γ)$ is the Laplace transform of a positive measure of total measure $ϕ^{an}(x, s_0) < \infty$, under certain conditions on the parameters governing the function. We deduce log-convexity of this function, as well as positive-definiteness and an integral formula for a certain intertwining operator associated with the Askey-Wilson operator $L$. This offers a $q$-analogue of several results, in the trigonometric limit due to Sahi-McSwiggen, and in the rational (i.e. Dunkl) limit due to Rösler.