A Convenient Alternative for Series Manipulation via the Translation Operator

TL;DR

This paper introduces a novel, efficient method for manipulating power series using the translation operator expressed as an infinite product, simplifying calculations in multivariate expansions and series operations.

Contribution

It presents a new technique that employs combinatorial arguments and infinite product representation of the translation operator for series manipulation, enhancing efficiency and applicability.

Findings

Effective for multivariate expansions

Simplifies operations like reciprocals and fractional powers of series

Demonstrated with electrostatic multipole expansion and special functions

Abstract

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