Related papers: On a finite trigonometric sum related to Dedekind …
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…
Two classes of finite trigonometric sums, each involving only $\sin$'s, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.
Several methods are used to evaluate finite trigonometric sums. In each case, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
Finite trigonometric Fourier series on a set of discrete equidistant points are considered. A finite system of orthogonal functions that have interpolation and certain differential properties on the period is introduced. Finite Fourier…
We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…
The finite sums of powers of cosecs occur in numerous situations, both physical and mathematical, examples being the Casimir effect, Renyi entropy, Verlinde's formula and Dedekind sums. I here present some further discussion which consists…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
In this paper we construct trigonometric functions of the sum T_{p}(h,k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric representations of this sum. We prove…
We give an explicit expression of the elliptic classical Dedekind sum which is a special case of multiple elliptic Dedekind sums introduced by Egami. We also determine the denominator of the rational part and zeros of the elliptic classical…
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a…
We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed…
I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of the angle allows for an…
We obtain new trigonometric identities, which are some product-to-sum type formulas for the higher derivative of the cotangent and cosecant functions. Further, from specializations of our formulas, we derive not only various known…
We define and study the interpolated finite multiple harmonic $q$-series. A generating function of the sums of the interpolated finite multiple harmonic $q$-series with fixed weight, depth and $i$-height is computed. Some Ohno-Zagier type…
Dedekind sums have applications in quite a number of fields of mathematics. Therefore, their distribution has found considerable interest. This article gives a survey of several aspects of the distribution of these sums. In particular, it…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…
We calculate some finite and infinite sums containing the digamma function in closed-form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative…
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits…