On quadratic Gauss sums and variations thereof (original) (raw)
On Closed Forms of Some Trigonometric Series
Axioms, 2024
We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative.
Closed-form summation of some trigonometric series
Mathematics of Computation, 1995
The problem of numerical evaluation of the classical trigonometric series \[ S ν ( α ) = ∑ k = 0 ∞ sin ( 2 k + 1 ) α ( 2 k + 1 ) ν and C ν ( α ) = ∑ k = 0 ∞ cos ( 2 k + 1 ) α ( 2 k + 1 ) ν , {S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},} \] where ν > 1 \nu > 1 in the case of S 2 n ( α ) {S_{2n}}(\alpha ) and C 2 n + 1 ( α ) {C_{2n + 1}}(\alpha ) with n = 1 , 2 , 3 , … n = 1,2,3, \ldots has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when α \alpha is equal to a rational multiple of 2 π 2\pi , these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving C ν ( α ) {C_\nu }(\alpha ) and S ν ( α ) {S_\nu }(\alpha ) in terms of the generalized Riemann zeta funct...
2021
This kind of character sum has been studied for a long time. The values of G(n, χ; q) behave irregularly whenever χ varies. For a positive integer n with gcd(n, q) = 1, one can find a non-trivial upper bound of |G(n, χ; q)|. For such results see the work of Cochrane and Zheng [3]. In case of prime p, finding such bounds is due to Weil [9]. Let p be an odd prime and L(s, χ) denote the Dirichlet L-function corresponding to the character χ mod p. Let χ0 denote the principal character modulo p. For a general integer m ≥ 3, whether there exists an asymptotic formula for
On the Summations of Some Special Sequences
2020
This paper is concerned with the summations of the Pell, Lucas, Pell-Lucas, Jacopsthal, Jacopsthal-Lucas, generalized Pell, generalized dual Pell and generalized dual Pell quaternion sequences. Also, summing infinite series of reciprocals of the Pell numbers is calculated.
On certain trigonometric sums in several variables
Collectanea Mathematica
The Hilbert space structure of totally even functions (mod r) which depend on extended Ramanujan sums is described. The function ε k defined as the quotient of Jordan's J k -function and Euler's φ-function is introduced as a new generalization of the Dedekind ψ-function. Using the basic methods of totally even functions (mod r), we point out that ε k has also a purpose to serve in obtaining the k-dimensional analogue of an identity due to P. Kesava Menon.
A Note on Two Results Contiguous to a Quadratic Transformation Due to Gauss with Applications
2019
The aim of this paper is to establish two new results contiguous to a well-known, interesting and very useful quadratic transformation due to Gauss. As applications, we first obtain two summation formulas for the series 3F2 with unit argument closely related to the classical Watson’s summation theorem and then two new identities closely related to that obtained by Krattenthaler and Rao (2003).