Recursive Calculation Method for the Second Kind of Associated Legendre Functions and Its First and Second Derivatives

This paper derives the ellipsoid harmonic series of earth's gravitational field about the second kind of associated Legendre functions and its first and second derivatives recursive calculation method, and it compares their results with those of traditional Jekeli recursive method. The result shows that the numbers with maxima of about 30 is required terms of the revised Jekeli's recurrences is about half numbers compared with the traditional recurrences, when the recursive computation has the same convergence accuracy. As degree n, order m and convergent k increase, the revised Jekeli's recurrences is all fulfilled to 1×10^-6. Therefore, we also can see that the relationship of the height h and degree n in the ellipsoid approximation, as similar with the approximation of spherical harmonics's (R/r)ⁿ⁺¹ in different heights.