Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions
type: Item
instance of: scholarly article
title: Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions
zbMATH Open document ID: 0860.33013
DOI: 10.1007/BF00047669
author: Ph. Feinsilver
published in: Acta Applicandae Mathematicae
publication date: 1996-08-05T00:00:00Z
review text: This paper starts with certain representations of the complexification of the Lie algebra of an Euclidean group where bases of the corresponding Hilbert spaces are given in terms of Lommel polynomials and Bessel functions. The author then interpretes certain elements of these Lie algebras as quantum random variables. The spectral forms of the associated selfadjoint operators then lead to classical random variables \(X_t\) on \(\mathbb{Z}\) whose distributions satisfy recurrence relations known from the Bessel functions \(J_n\). The distributions of these random variables are explicitly given by \[ P(X_t=\pm m)=J_m(t)^2 \qquad (m\in\mathbb{Z},\;t\in\mathbb{R}) \] and called Lommel distributions. The remaining parts of the paper contain properties of these Lommel distributions: Generating functions, expectations, convergence of \(X_t/t\) in distribution to an inverse cosine distribution for \(t\to\infty\), a further central limit theorem involving Airy functions as limit, and finally some exponential bounds. Moreover, applications to electronic music and quantum mechanics on one-dimensional lattices are mentioned.
Mathematics Subject Classification ID: 33D80
Mathematics Subject Classification ID: 17D99
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 81R05
zbMATH DE Number: 885580
zbMATH Keywords: Lommel polynomials
zbMATH Keywords: quantum probability
zbMATH Keywords: Euclidean group
zbMATH Keywords: Bessel functions
zbMATH Keywords: limit theorem
zbMATH Keywords: Airy functions
Wikidata QID: Q115395289
reviewed by: Michael Voit
MaRDI profile type: Publication
cites work: Q3331506
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description: scientific article; zbMATH DE number 885580