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.


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