Recursive computation of oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives

A recursive method is developed to compute the ratios of the oblate spheroidal harmonics of the second kind and their first-, second-, and third-order derivatives. The recurrence formulas consist of three kinds: (1) fixed-degree increasing-order, (2) mixed-degree increasing-order, and (3) fixed-order decreasing-degree. The three seed values are evaluated by rapidly convergent series. The derivatives of the ratios are recursively obtained from the values and lower-order derivatives of the same harmonic order and of the same or higher degrees. The new method precisely and quickly computes the ratios and their low-order derivatives. It provides 13 correct digits of the ratios of degree as high as 262,000 and runs 20–100 times faster than the existing methods.