Analytical equations for an infinite series involving low-order associated Legendre functions in geoscience

The associated Legendre functions constituting the kernel function of spherical harmonics have a wide range of applications in geodesic and geophysical fields, such as calculating the Green’s functions for a spherical Earth model. The analytical expressions for the infinite series involving the associated Legendre functions are useful. In this paper, starting with the generating function, we present a set of analytical equations for an infinite series involving associated low-order m=0,1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {m = 0,~1,~2} \right)$$\end{document} Legendre functions. After careful verification, the accuracy and effectiveness of the nearly sixty listed equations are confirmed. The open-source code written using the Wolfram language, GNU octave/MATLAB, and Fortran-90 are available through GitHub (https://github.com/UCAStanghe2014/analytical_sums_associated_Legendre).