Moments of q-Normal and conditional q-Normal distributions

TL;DR

This paper derives moments and moment generating functions for q-Normal and conditional q-Normal distributions, which generalize classical distributions like Normal, Wigner, and Kesten-McKay, and explores related asymptotic properties.

Contribution

It provides explicit calculations of moments and generating functions for these generalized distributions, extending understanding of their properties.

Findings

Derived moments and generating functions for q-Normal distributions

Connected q-Normal and conditional q-Normal to classical distributions

Analyzed asymptotic properties of Bessel function expansions

Abstract

We calculate moments and moment generating functions of two distributions: the so called $q-$Normal and the so called conditional $q-$Normal distributions. These distributions generalize both Normal ($q=1),$ Wigner ($% q=0,$ $q-$Normal) and Kesten-McKay ($q=0,$ conditional $q-$Normal) distributions. As a by product we get asymptotic properties of some expansions in modified Bessel functions.