Translation matrix elements for spherical Gauss–Laguerre basis functions

Spherical Gauss–Laguerre (SGL) basis functions, i.e., normalized functions of the type Ln-l-1(l+1/2)(r2)rlYlm(ϑ,φ),|m|≤l<n∈N, constitute an orthonormal polynomial basis of the space L ² on 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.