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 \(\left( {m = 0,~1,~2} \right)\) 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).

构成球谐函数核函数的相关勒让德函数在测地线和地球物理领域具有广泛的应用，例如计算球形地球模型的格林函数。涉及关联勒让德函数的无限级数的解析表达式很有用。在本文中，从生成函数开始，我们提出了一组涉及相关低阶的无穷级数的解析方程\(\left( {m = 0,~1,~2} \right)\)勒让德函数。经过仔细验证，证实了列出的近六十个方程的准确性和有效性。使用 Wolfram 语言、GNU Octave/MATLAB 和 Fortran-90 编写的开源代码可通过 GitHub (https://github.com/UCAStanghe2014/analytical_sums_related_Legendre) 获得。