Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations

Abstract

The computation of the Whittaker function Wκ,μ(z) via its divergent asymptotic expansion is discussed. This divergent series can be summed with the help of Padé approximants, which convert its partial sums into rational functions [C. de Izarra, O. Vallée, J. Picart, and N. Tran Minh, Comput. Phys. 9, 318 (1995)]. However, this and related divergent series, as they for instance occur in special function theory or in quantum mechanical perturbation theory, can be summed much more efficiently by some alternative rational approximants that use explicit estimates for the truncation error [D. Levin, Int. J. Comput. Math. B 3, 371 (1973); E.J. Weniger, Comput. Phys. Rep. 10, 189 (1989)]. © 1996 American Institute of Physics.