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Chebshev polynomials of the first kind are applied to telescope both the far-field multipole expansions and the near-field Taylor series expansions used in solving large N-body problems via fast multipole methods. The technique is demonstrated for pairwise-additive, 1/r interparticle potentials in Cartesian coordinates, and a general Mathematica package is provided to derive the modified expansion coefficients symbolically. Accelerated convergence and more uniform error can be achieved without additional computations during runtimes. Hence the telescoped series require fewer expansion terms for a given accuracy requirement, saving considerable computational expense over conventional fast multipole implementations.
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