Free Exponential Families as Kernel Families

Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel instead of the exponential kernel. We use this approach to re-derive several known results and to study further similarities with exponential families and reproductive exponential models.