Algorithms for Rational Approximations for a Confluent Hypergeometric Function.

摘要

In the authors' volumes on the special functions, rational approximations for the confluent hypergeometric function (z sup a)U(a;c;z) were examined in some detail. This confluent function is very important in the applications since it includes as special cases the incomplete gamma function (special cases of which are exponential, sine and cosine integrals, Fresnel integrals and the error 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. FORTRAN programs are provided to evaluate the Pade and non-Pade rational approximations by using the appropriate recursion formulas to generate the numerator and denominator polynomials as a number, and to also evaluate the coefficients which define these polynomials.