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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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