Some Relations Between the Riemann Zeta Function and the Generalized Bernoulli Polynomials of Level $m$

The main purpose of this paper is to show some relations between the Riemann zeta function and the generalized Bernoulli polynomials of level $m$. Our approach is based on the use of Fourier expansions for the periodic generalized Bernoulli functions of level $m$, as well as quadrature formulae of Euler-Maclaurin type. Some illustrative examples involving such relations are also given.

Keywords:

Bernoulli polynomials, Euler-Maclaurin quadrature formulae, generalized Bernoulli polynomials of level $m$ quadrature formula, Riemann zeta function,

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