On the Gaussian q-Distribution

We present a study of the Gaussian q-measure introduced by Diaz and Teruel from a probabilistic and from a combinatorial viewpoint. A main motivation for the introduction of the Gaussian q-measure is that its moments are exactly the q-analogues of the double factorial numbers. We show that the Gaussian q-measure interpolates between the uniform measure on the interval [-1,1] and the Gaussian measure on the real line.