This note explains how the two measures used to define the µ-deformed Segal-Bargmann space are natural and essentially unique structures. As is well known, the density with respect to Lebesgue measure of each of these measures involves a Macdonald function. Our primary result is that these densities are the solution of a system of ordinary differential equations which is naturally associated with this theory. We then solve this system and find the known densities as well as a “spurious ” solution which only leads to a trivial holomorphic Hilbert space. This explains how the Macdonald functions arise in this theory. Also we comment on why it is plausible that only one measure will not work. We follow Bargmann’s approach by imposing a condition sufficient for the µ-deformed creation and annihilation operators to be adjoints of each other. While this note uses elementary techniques, it reveals in a new way basic aspects of the structure of the µ-deformed Segal-Bargmann space. Keywords: Segal-Bargmann analysis, µ-deformed quantum mechanics. 1

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal-Bargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Saïd and Ørsted and independently by Soltani) and the Dunkl heat kernel, due to Rösler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal-Bargmann analysis on Euclidean space. Hall’s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in µ-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal-Bargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall’s Version C generalized Segal-Bargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal-Bargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal-Bargmann kernels for