Translation matrix elements for spherical Gauss-Laguerre basis functions

Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type $L_{n-l-1}^{(l+1\mathrm{/}2)}(r^2)r^l{Y}_{lm}(\vartheta ,\phi )$, $\mathrm{\mid }m\mathrm{\mid }\le l<n\in \mathbb{N}$, 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.