On a finite sum with powers of cosines

Abstract

A formula for the finite sum of powers of cosines with fractional multiples of ?/2, viz. S(n,m) = ?m, k=1 (-1)k cos2n (k? / (2m + 2)), where m and n are arbitrary positive integers, is derived. In the process new and interesting mathematical results are uncovered, particularly with regard to the Bernoulli and Euler polynomials, while other related series are discussed. It is found that the series always yields rational values, which can only be evaluated by using the integer arithmetic routines in a mathematical software package such as Mathematica.