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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 \(\cos\).

MSC:
11B68 Bernoulli and Euler numbers and polynomials
42A05 Trigonometric polynomials, inequalities, extremal problems
11L03 Trigonometric and exponential sums (general theory)
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References:
[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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