On the Gaussian q-distribution

We present a study of the Gaussian q-measure introduced by Díaz 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.