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) 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).

Automated summary

This study presents a set of analytical equations for an infinite series involving associated low-order Legendre functions, based on the generating function, with nearly sixty equations carefully verified and confirmed for accuracy and effectiveness. The open-source code written in Wolfram language, GNU Octave/MATLAB, and Fortran-90 is available on GitHub.