On the Gaussian q-Distribution

TL;DR

This paper explores the Gaussian q-measure, connecting it to probabilistic and combinatorial perspectives, and demonstrates its role as an interpolation between uniform and Gaussian measures, with moments linked to q-analogues of double factorials.

Contribution

It provides a new probabilistic and combinatorial analysis of the Gaussian q-measure, highlighting its interpolation properties and moment structure.

Findings

Gaussian q-measure interpolates between uniform and Gaussian measures

Moments of the measure are q-analogues of double factorials

Offers new insights into q-analogues in probability and combinatorics

Abstract

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.