引用本文: 魏子卿. 完全正常化缔合勒让德函数及其导数与积分的递推关系[J]. 武汉大学学报 ( 信息科学版), 2016, 41(1): 27-36.
WEI Ziqing. Recurrence Relations for Fully Normalized Associated Legendre Functions and Their Derivatives and Integrals[J]. Geomatics and Information Science of Wuhan University, 2016, 41(1): 27-36.
 Citation: WEI Ziqing. Recurrence Relations for Fully Normalized Associated Legendre Functions and Their Derivatives and Integrals[J]. Geomatics and Information Science of Wuhan University, 2016, 41(1): 27-36.

## Recurrence Relations for Fully Normalized Associated Legendre Functions and Their Derivatives and Integrals

• 摘要: 在地球重力场问题中,常用到完全正常化缔合勒让德函数及其导数、积分的递推关系。当前流行的地球扰动位模型均采用完全正常化的缔合勒让德函数,用此类模型可以高效方便计算各种扰动重力场元。随着本世纪多个新一代卫星重力探测计划成功实施,高阶或超高阶地球重力场模型的研究备受学界的关注。有关完全正常化缔合勒让德函数的递推关系对于高阶重力场模型具有特别意义。本文在前人研究的基础上,用初等微积分导出了若干新的递推关系式。同时还推导了正常化缔合勒让德函数及其导数、积分的检核式,这些检核式涉及地球位的球谐级数的数学性质。

Abstract: Recurrence relations for fully normalized associated Legendre functions and their derivatives and integrals are often used when studying the Earth gravity field. Fully normalized associated Legendre functions are all adopted in current popular Earth disturbance potential models make the calculations of various disturbance gravity field elements highly efficient and easy when such models and related recurrence relations are used. New generation satellite gravity exploration missions have been successfully implemented, and high or ultrahigh degree Earth gravity field models have been quickly developed. This study area has received much attention in academic community; our investigation has a special meaning as it addresses the recurrence relations of fully normalized associated Legendre functions. Building upon previous research, several new recurrence relation expressions are derived in detail based on elementary calculus as the mathematical tool for the derivation. Formulas for assessing the values of fully normalized associated Legendre functions, their derivatives, and integrals are also deduced. This study involves the mathematical properties of the spherical harmonic expansion series of the geo-potential, thus is related to fundamental theory research.

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