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On a finite sum with powers of cosines

da Fonseca C.M. (Universidade de Coimbra, Departamento de Matematica, Coimbra, Portugal)
Kowalenko Victor (The University of Melbourne, Department of Mathematics and Statistics, ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Victoria, Australia)

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.

Keywords: algorithm, Bernoulli polynomial, coefficient, correction term, cosecant number, cosecant polynomial, cosine, cosine series, cyclic, Euler polynomial, finite sum, power, program, rational, secant number, secant polynomial, trigonometric