An approximation theoretic alternative to asymptotic expansions for special functions

Abstract

Some classes of functions, which are solutions of ordinary linear homogeneous differential equations of second order with an irregular singularity at infinity, posses asymptotic expansions with respect to a real positive variable at infinity. In the case of non-oscillatory behaviour of such functions these asymptotic expansions can be replaced by near-optimal relative approximations by polynomials of the reciprocal variable and by approximations with rational functions, using the so-called Carathéodory-Fejér method. The investigations include Kummer functions resp. Whittaker functions (confluent hypergeometric functions) with this behaviour. A large class of special functions can be considered as Kummer functions resp. Whittaker functions. Two examples concerning the incomplete Γ function and the transformed Gaussian probability function are given in some detail.