On the hybrid power mean of two kind different trigonometric sums

On Some Trigonometric Power Sums

In contrast to Fourier series, these finite power sums are over the angles equally dividing the upper-half plane. Moreover, these beautiful and somewhat surprising sums often arise in analysis. In this note, we extend the above results to the power sums as shown in identities (17), (19), (25), (26), (32), (33), (34), (35), and (36) and in the appendix. The method is based on the generating func...

On the Hybrid Mean Value of Cochrane Sums and Two-term Exponential Sums

Basic Trigonometric Power Sums with Applications

− 2 , where m is a positive integer and subject to m < n + 1 [28]. In this equation ⌊n/2⌋ denotes the floor function of n/2 or the greatest integer less than or equal to n/2. A solution to the above problem was presented shortly after by Greening et al. in [18]. Soon afterwards, there appeared a problem involving powers of the secant proposed by Gardner [15], which was solved partially by Fishe...

A hybrid mean value involving a new Gauss sums and Dedekind sums

‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.

On Trigonometric Sums with Gaps

A well known theorem states as follows :' Let ni < n2 <. . ., nk+1 / nk > A > 1 be an infinite sequence of real numbers and S (ak + bk) a divergent series satisfying k=1 Then denotes the Lebesgue measure of the set in question. It seems likely that the Theorem remains true if it is not assumed that the n k are integers. On the other hand if nk ,f-n,.-1 is an arbitrary sequence of integers it is...