An integral approach to the Gardner-Fisher and untwisted Dowker sums

We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$S_{m,v}={\left(\frac{\pi }{2m}\right)}^{2v}\sum _{k=1}^{m-1}{\cos }^{-2v}\left(\frac{k\pi }{2m}\right),$$ We present a new and elegant integral approach to computing the Gardner-Fisher trigonometric power sum, which is given by $$S_{m,v}={\left(\frac{\pi }{2m}\right)}^{2v}\sum _{k=1}^{m-1}{\cos }^{-2v}\left(\frac{k\pi }{2m}\right),$$ where $m$ and $v$ are positive integers. This method not only confirms the results obtained earlier by an empirical method, but it is also much more expedient from a computational point of view. By comparing the formulas from both methods, we derive several new interesting number theoretic results involving symmetric polynomials over the set of quadratic powers up to $(v-1{)}^2$ and the generalized cosecant numbers. The method is then extended to other related trigonometric power sums including the untwisted Dowker sum. By comparing both forms for this important sum, we derive new formulas for specific values of the N"{o}rlund polynomials. Finally, by using the results appearing in the tables, we consider more advanced sums involving the product of powers of cotangent and tangent with powers of cosecant and secant respectively.