Moments of q-Normal and conditional q-Normal distributions

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 ( $q-$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.

References in corpus (7)

• $q$-Gaussian processes: non-commutative and classical aspects
• The q-harmonic oscillator and the Al-Salam and Carlitz polynomials
• Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)
• Random matrices, non-backtracking walks, and orthogonal polynomials
• $q-$Wiener ($\alpha ,q)-$ Ornstein-Uhlenbeck processes. A generalization of known processes
• Befriending Askey-Wilson polynomials
• A few remarks on quadratic harnesses

Cited by in corpus (7)

• Asymmetric Simple Exclusion Process with open boundaries and Quadratic Harnesses
• Bivariate $q$-normal distribution for transition strengths distribution from many-particle random matrix ensembles generated by $k$-body interactions
• Wavefunction structure in quantum many-fermion systems with $k$-body interactions: conditional $q$-normal form of strength functions
• Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials
• On the generalized Kesten--McKay distributions
• Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions
• Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths