Summary: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind and their first derivatives with respect to are shown to provide accurate values, even for low values of where the traditional expressions provide increasingly inaccurate results as the size parameter 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 and the prolate spheroidal angular function of the first kind in a series of products of the corresponding spherical functions.
B. J. King and A. L. van Buren [SIAM J. Math. Anal. 4, 149-160 (1973; Zbl 0249.33011)] 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 and its first derivative.