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

DOI:
https://doi.org/10.1090/S0002-9939-1995-1283544-0

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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Additional Information

Keywords:
Bernoulli polynomials,
Euler polynomials,
Hurwitz zeta function,
summation of series

Article copyright:
© Copyright 1995
American Mathematical Society