Some classes of functions, which are solutions of ordinary linear homogeneous differential equations of second order with an irregular singularity at infinity possess asymptotic expansions with respect to a real positive variable at infinity. In the case of non-oscillatory behavior of such functions these asymptotic expansions can be replaced by near-best relative approximations by polynomials of the reciprocal variable and by approximations with rational functions, using the socalled Carathéodory-Féjèr 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 Gamma function and the transformed Gaussian probability function are given in some detail.
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