Exact formulas for additional terms in some 1mportant series expansions

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Abstract
An investigation into previous work dealing with various series expansions about the normal distribution revealed little in the way of exact formulas for the terms in these expansions. In fact, for the univariate case, no more than the first few terms in any of the expansions have been published, with the exception of the Cornish-Fisher expansions, for which the first six terms are given in the Cornish-Fisher (1960) paper. In order to use the expansions in other problems, we first needed the exact formulas for some higher order terms. This paper is devoted to the presentation of these new terms and to the techniques used to obtain them. After this work was begun, a paper by Hill and Davis (1968) dealt with the theory of the same problem and some unpublished tables were referenced. The present paper, based one equivalent theory, but done independently using different computing algorithms, provides exact formulas for, at least, the first eight terms in four important expansions. The unpublished results of Hill and Davis were found to be correct to the order here computed (terms to order n−4) . Higher order results were not computed because of limitations on computing funds for this project; no particular difficulty is involved in computing higher order terms.

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