Formulas express specific sums of 1/sin^s at dyadic rational angles via generalized Bernoulli and Euler polynomials, yielding an integral form for zeta(s).
Sums of Powers of Sine and Generalized Bernoulli Polynomials
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abstract
We produce formulas for $$\sum_{j=1}^{2^{n-2}}\frac{1}{\sin^s\left(\frac{(2j-1)\pi}{2^n}\right)}$$ in terms of Generalized Bernoulli and Euler polynomials and use one of the formulas to produce a nice integral representation of the Riemann zeta function.
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Sums of Powers of Sine and Generalized Bernoulli Polynomials
Formulas express specific sums of 1/sin^s at dyadic rational angles via generalized Bernoulli and Euler polynomials, yielding an integral form for zeta(s).