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

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.