On the Hybrid Power Mean of Two-Term Exponential Sums and Cubic Gauss Sums

In this paper, an interesting third-order linear recurrence formula is presented by using elementary and analytic methods. This formula is concerned with the calculating problem of the hybrid power mean of a certain two-term exponential sums and the cubic Gauss sums. As an application of this result, some exact computational formulas for one kind hybrid power mean of trigonometric sums are obtained.

Another two families of integer-valued polynomials associated with finite trigonometric sums

As a sequel to our recent paper, its general approach was here extended to finite alternating trigonometric sums giving rise to polynomials which were systematically examined in full detail as well as in a unified manner using simple arguments. Two new general families of integer-valued polynomials (along with four other families derived from them, also integer{valued, including two already known) were deduced. Also, these polynomials enable closed-form summation of a great deal of general families of finite sums.

On the Study of Trigonometric Polynomials Using Strum Sequence

This article constructs trigonometric polynomials of the sine and cosine whose sums are nonnegative. As an application, those nonnegative trigonometric sums are used to study the geometric properties of complex polynomials in the unit disk. The Strum sequences are used to prove the main outcome.