In this paper, we discuss some methods for the calculation of the Legendre polynomial difference \(P_{n - 1}(t) - P_{n + 1}(t)\). Such differences frequently appear when calculating the spherical harmonic coefficients of truncated spatial filters. By examining the relation between the Legendre polynomials and other classical orthogonal polynomials (i.e., Gegenbauer and Jacobi polynomials), we provide analytical expressions for \(P_{n - 1}(t) - P_{n + 1}(t)\) in terms of the latter. We also derive a recurrence relation that avoids the direct evaluation of the Legendre polynomials. Analogous analytical expressions and recurrence relations are provided for the calculation of the \(P_{n - 1}(t) - P_{n + 1}(t)\) derivative and integral. We show that these recurrence relations can be derived from a more general recurrence relation; thus, the former are special cases of the latter. The expressions given in this study can be useful for the more efficient calculation of truncated kernels in the spherical harmonic domain. We demonstrate the applicability of these new expressions for the Pellinen (i.e., rectangular) filter kernel that is regularly used in regional gravity field modeling.

在本文中，我们讨论了计算勒让德多项式差分\(P_{n - 1}(t) - P_{n + 1}(t)\) 的一些方法。这种差异在计算截断空间滤波器的球谐系数时经常出现。通过检查勒让德多项式与其他经典正交多项式（即 Gegenbauer 和 Jacobi 多项式）之间的关系，我们提供了\(P_{n - 1}(t) - P_{n + 1}(t)\) 的解析表达式就后者而言。我们还推导了避免对勒让德多项式进行直接评估的递归关系。为计算\(P_{n - 1}(t) - P_{n + 1}(t)\)提供了类似的解析表达式和递推关系导数和积分。我们表明这些递推关系可以从更一般的递推关系导出；因此，前者是后者的特例。本研究中给出的表达式可用于更有效地计算球谐域中的截断核。我们证明了这些新表达式对于区域重力场建模中经常使用的 Pellinen（即矩形）滤波器核的适用性。