On quadratic Gauss sums and variations thereof (original) (raw)
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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...
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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 Sums of Certain Classes of Series
Communications of the Korean Mathematical Society, 2012
The aim of this research note is to provide the sums of the series ∞ k=0 (−1) k a − i k 1 2 k (a + k + 1) for i = 0, ±1, ±2, ±3, ±4, ±5. The results are obtained with the help of generalization of Bailey's summation theorem on the sum of a 2 F 1 obtained earlier by Lavoie et al.. Several interesting results including those obtained earlier by Srivastava, Vowe and Seiffert, follow special cases of our main findings. The results derived in this research note are simple, interesting, easily established and (potentially) useful.
2022
The aim of this paper is to give some new classes of finite sums involving the numbers y (m, λ), the generalized harmonic functions, special numbers and polynomials, the Dedekind sums, and other combinatorial sum. Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.