EXPANSIONS FOR SOME CONFLUENT HYPERGEOMETRIC-FUNCTIONS

Power series for the parabolic cylinder function, D(v)(z), are derived from known addition theorems, thus enabling a more compact and efficient expression of the function. One of the expansions is applied to find an asymptotic approximation for the function as absolute value of v --> infinity with \arg(-v)\ < pi. Next, a large-argument (z) asymptotic addition theorem is derived from an integral representation of D(v)(z). Before extending this type of summation theorem to the two general confluent hypergeometric functions, motivating mathematical applications to three integration problems are given as examples of the practical usefulness of the formulae.