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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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