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New formulae for the Bernoulli and Euler polynomials at rational arguments

Authors: Djurdje Cvijović and Jacek Klinowski
Journal: Proc. Amer. Math. Soc. 123 (1995), 1527-1535
MSC: Primary 11M35; Secondary 11B68, 33E99
MathSciNet review: 1283544
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Abstract: We prove theorems on the values of the Bernoulli polynomials ${B_n}(x)$ with $n = 2,3,4, \ldots$, and the Euler polynomials ${E_n}(x)$ with $n = 1,2,3, \ldots$ for $0 \leq x \leq 1$ 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 $\zeta (z,\alpha )$. 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.

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Keywords: Bernoulli polynomials, Euler polynomials, Hurwitz zeta function, summation of series
Article copyright: © Copyright 1995 American Mathematical Society