# Translation matrix elements for spherical Gauss-Laguerre basis functions

## TL;DR

This paper derives a closed-form expression for translation matrix elements of spherical Gauss-Laguerre basis functions, enabling efficient computation crucial for 3D rigid matching problems.

## Contribution

We present a novel closed-form formula for SGL translation matrix elements, facilitating practical and efficient spectral analysis under translations.

## Key findings

- Derived a closed-form expression for translation matrix elements.
- Enables direct and efficient computation of spectral translation behavior.
- Supports improved algorithms for 3D rigid matching tasks.

## 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.