Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions
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.
scientific article; zbMATH DE number 885580