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New Formulae for the Bernoulli and Euler Polynomials at Rational Arguments

Djurdje Cvijović and Jacek Klinowski
Proceedings of the American Mathematical Society
Vol. 123, No. 5 (May, 1995), pp. 1527-1535
DOI: 10.2307/2161144
Stable URL: http://www.jstor.org/stable/2161144
Page Count: 9
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New Formulae for the Bernoulli and Euler Polynomials at Rational Arguments
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Abstract

We prove theorems on the values of the Bernoulli polynomials Bn(x) with n = 2, 3, 4, ..., and the Euler polynomials En(x) with n = 1, 2, 3, ... for 0 ≤ x ≤ 1 where x is rational. Bn(x) and En(x) are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function ζ(z, α). The well-known argument-addition formulae and reflection property of Bn(x) and En(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.

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