Complementary moments and the summation of series

Abstract

In previous papers the author has treated the summation of power and other series whose coefficients re moments of a function f( u), usually over the interval 0 </ u </ 1. The present paper defines “complementary moments” of f( u) for the same interval, and demonstrates their use in convergence acceleration and in the analytic summation of series having somewhat complicated coefficients of a rather special form. Simple algorithms are established, and these enable one, for example, to obtain exact Laplace-transform inverses of unusual functions of the Laplace-transform operator p.