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We derive and discuss a technique for manipulating power series which iscomplementary to standard procedures. We begin with the translation operator,but we express the operator as an infinite product instead of expanding it as aseries and we apply combinatorial arguments to generate the terms in the seriesin an efficient manner with a minimum of clutter and intermediate calculations.The method is effective for developing multivariate expansions, and may also beused to manipulate series, e.g. in operations where one must take thereciprocal of a power series or raise it to a power that may be fractional orirrational. In the case of two component perturbations, we obtain analyticexpressions for the expansion coefficients. We use our technique to generate anelectrostatic multipole expansion as a demonstration of its utility inproducing coefficients of special functions such as the Legendre and Hermitepolynomials.

A Convenient Alternative for Series Manipulation via the Translation Operator