Title: Translation matrix elements for spherical Gauss-Laguerre basis functions,
Abstract: Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of
the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta,\varphi)$, $|m|
\leq l < n \in \mathbb{N}$, constitute an orthonormal polynomial basis of the
space $L^{2}$ on $\mathbb{R}^{3}$ with radial Gaussian weight $\exp(-r^{2})$.
We have recently described reliable fast Fourier transforms for the SGL basis
functions. The main application of the SGL basis functions and our fast
algorithms is in solving certain three-dimensional rigid matching problems,
where the center is prioritized over the periphery. For this purpose, so-called
SGL translation matrix elements are required, which describe the spectral
behavior of the SGL basis functions under translations. In this paper, we
derive a closed-form expression of these translation matrix elements, allowing
for a direct computation of these quantities in practice.