Algorithms for Various Trigonometric Power Sums

In this paper, algorithms for different types of trigonometric power sums are developed and presented. Although interesting in their own right, these trigonometric power sums arise during the creation of an algorithm for the four types of twisted trigonometric power sums defined in the introduction. The primary aim in evaluating these sums is to obtain exact results in a rational form, as opposed to standard or direct evaluation, which often results in machine-dependent decimal values that can be affected by round-off errors. Moreover, since the variable, m, appearing in the denominators of the arguments of the trigonometric functions in these sums, can remain algebraic in the algorithms/codes, one can also obtain polynomial solutions in powers of m and the variable r that appears in the cosine factor accompanying the trigonometric power. The degrees of these polynomials are found to be dependent upon v, the value of the trigonometric power in the sum, which must always be specified.