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Bernoulli numbers and polynomials of arbitrary complex indices. (English) Zbl 0768.11010
For complex $\alpha$ with $\text{Re }\alpha > 1$ the authors define the Bernoulli periodic function ${\cal B}\sb \alpha(x)$ with period 1 by the Fourier series $${\cal B}\sb \alpha = -2\Gamma(\alpha + 1)\sum\sp \infty\sb{k=1}{\cos(2\pi kx - \alpha\pi/2)\over (2\pi k)\sp \alpha},$$ and study its connection with the classical Bernoulli polynomials and Bernoulli numbers.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials
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##### References:
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