Multisection of series

Extract

In a recent paper [1], the authors gave a combinatorial interpretation to sums of equally spaced binomial coefficients. Others have been interested in finding such sums, known as multisection of series. For example, Gould [2] derived interesting formulas but much of his work involved complicated manipulations of series. When the combinatorial approach can be implemented, it is neat and efficient. In this paper, we present another approach for finding equally spaced sums. We consider both infinite sums and partial finite sums based on generating functions and extracting coefficients.

While generating functions were first introduced by Abraham de Moivre at the end of seventeen century, its systematic use in combinatorial analysis was inspired by Leonhard Euler. Generating functions got a new birth in the twentieth century as a part of symbolic methods. As a central mathematical tool in discrete mathematics, generating functions are an essential part of the curriculum in the analysis of algorithms [3, 4]. They provide a bridge between discrete and continuous mathematics, as illustrated by the fact that the generating functions presented here appear as solutions to corresponding differential equations.