Bernoulli numbers and polynomials of arbitrary complex indices (Q1206222)

For complex \(\alpha\) with \(\text{Re }\alpha > 1\) the authors define the Bernoulli periodic function \({\mathcal B}_ \alpha(x)\) with period 1 by the Fourier series \[ {\mathcal B}_ \alpha = -2\Gamma(\alpha + 1)\sum^ \infty_{k=1}{\cos(2\pi kx - \alpha\pi/2)\over (2\pi k)^ \alpha}, \] and study its connection with the classical Bernoulli polynomials and Bernoulli numbers.