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 $R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )$ and their first derivatives with respect to $\xi$ are shown to provide accurate values, even for low values of $l - m$ 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 $R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )$ and the prolate spheroidal angular function of the first kind $S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right )$ 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 $R_{ml}^{\left ( 1 \right )}\left ( c, \xi \right )$ and its first derivative.