## New formulae for the Bernoulli and Euler polynomials at rational arguments.(English)Zbl 0827.11012

Authors’ summary: We prove theorems on the values of the Bernoulli polynomials $$B_n (x)$$ with $$n=2, 3,\dots$$ and the Euler polynomials $$E_n (x)$$ with $$n=2, 3,\dots$$ for $$0<x <1$$ where $$x$$ is rational. $$B_n (x)$$ and $$E_n (x)$$ are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function $$\zeta (z,a)$$. The well known argument addition formulae and reflection property of $$B_n (x)$$ and $$E_n (x)$$ extend this result to any rational argument. We also deduce new relations concerning the finite sums of the Hurwitz zeta function and sum some classical trigonometric series.

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials 33E99 Other special functions
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