  Using the technique developed in approximation theory, we construct examples
of exponential families of infinitely divisible laws which can be viewed as
deformations of the normal, gamma, and Poisson exponential families. Replacing
the differential equation of approximation theory by a q-differential equation,
we define the q-exponential families, and we identify all q-exponential
families with quadratic variance functions when |q|<1. We elaborate on the case
of q=0 which is related to free convolution of measures. We conclude by
considering briefly the case q>1, and other related generalizations.
