Algorithms for Rational Approximations for a Confluent Hypergeometric Function II.

Abstract : This is a sequel to a previous paper where rational approximations for the confluent hypergeometric function were treated. Here we take up rational approximations for (1)F(1)(a;c;-z). The confluent functions are very important in the applications since they include as special cases the incomplete gamma function Bessel functions, parabolic cylinder functions and Coulomb wave functions. In the special case where a is unity, the confluent function becomes an incomplete gamma function. In this event, complete a priori error analyses for the main diagonal Pade approximations and much more were presented. For general parameters, the rational approximations treated were not of the Pade class. It was shown that the rational approximations converge, but a complete a priori analysis was not available. One of the purposes of this report is to correct this deficiency.