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Evaluation of two trigonometric sums. (English) Zbl 0820.11010
Eisenstein’s 1844 proof of the law of quadratic reciprocity is based on the trigonometric identity \[ \sum_{k=1}^{m-1} \sin {{2k \alpha\pi} \over m} \cot {{k\pi} \over m} =m- 2\alpha, \] where \(\alpha\) and \(m\) are integers with \(0<\alpha <m\). The authors replace \(\cot\) by \(\cot^ n\) and evaluate the sums in terms of Bernoulli polynomials. They obtain similar evaluation formulas with \(\sin\) replaced by \(\cos\).

11B68 Bernoulli and Euler numbers and polynomials
42A05 Trigonometric polynomials, inequalities, extremal problems
11L03 Trigonometric and exponential sums (general theory)
Full Text: EuDML
[1] Handbook of Mathematical Functions. (M. Abramowitz and I. A. Stegun, Nat. Bur. Standards, Washington, D.C, 1964. · Zbl 0171.38503
[2] EISENSTEIN G.: Aufgaben und Lehrsatze. J. Reine Angew. Math. 27 (1844), 281-283. (In: Mathematische Werke. Band I (Second edition)) · ERAM 027.0797cj
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