Computation of the Regular Confluent Hypergeometric Function

A procedure, alternative to Hypergeometric1F1, for the computation of the regular confluent hypergeometric function 1F1(a; b; z) is suggested. The procedure, based on an expansion of the Whittaker function in series of Bessel functions, proves to be useful for large values of ‚az‚, whenever ‚z‚ is smaller than or comparable to 1. The numerical values of the confluent hypergeometric (Kummer) function, 1F1(a; b; z) or M(a, b, z), obtained by Hypergeometric1F1[a,b,z] are correct for moderate values of the parameters a and b and the variable z. However, if the parameter a is large, the function loses accuracy and computation time increases, unless the variable z is small enough that ‚az‚ is less than or comparable to 1. In this note, we propose a procedure to evaluate 1F1(a; b; z) when the values of the parameters and the variable are unfavorable for using Hypergeometric1F1[a,b,z]. The procedure is based on an expansion, given by Buchholz [Buchholz 1969, sec. 7.4], of the Whittaker function in terms of Bessel functions. The Kummer function, which is closely related to the Whittaker function, has an expansion