Summary: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind $R^{(1)}_{ml}(c,\xi)$ 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 $R^{(1)}_{ml}(c,\xi)$ and the prolate spheroidal angular function of the first kind $S^{(1)}_{ml}(c,\eta)$ in a series of products of the corresponding spherical functions.
{\it B. J. King} and {\it 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 $R^{(1)}_{ml}(c,\xi)$ and its first derivative.