Accurate Calculation of Prolate Spheroidal Radial Functions of the First Kind and Their First Derivatives

Alternative expressions for calculating the prolate spheroidal radial functions of the first kind R^] (c, £) and their first derivatives with respect to £ are shown to provide accurate values, even for low values of I — m where the traditional expressions provide increasingly inaccurate results as the size parameter c increases to large values. These expressions also converge in fewer terms than the traditional ones. They are obtained from the expansion of the product of R^n) (c, £) and the prolate spheroidal angular function of the first kind 5^ (c, rj) in a series of products of the corresponding spherical functions. King and Van Buren [12] had used this expansion previously in the derivation of a general addition theorem for spheroidal wave functions. The improvement in accuracy and convergence using the alternative expressions is quantified and discussed. Also, a method is described that avoids computer overflow and underflow problems in calculating i?^(c, £) and its first derivative.