*(English)*Zbl 1031.65034

Summary: Alternative expressions for calculating the prolate spheroidal radial functions of the first kind ${R}_{ml}^{\left(1\right)}(c,\xi )$ 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 ${R}_{ml}^{\left(1\right)}(c,\xi )$ and the prolate spheroidal angular function of the first kind ${S}_{ml}^{\left(1\right)}(c,\eta )$ 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 ${R}_{ml}^{\left(1\right)}(c,\xi )$ and its first derivative.

##### MSC:

65D20 | Computation of special functions, construction of tables |

33E10 | LamĂ©, Mathieu, and spheroidal wave functions |

33F05 | Numerical approximation and evaluation of special functions |