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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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New formulae for the Bernoulli and Euler polynomials at rational arguments
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by Djurdje Cvijović and Jacek Klinowski PDF
Proc. Amer. Math. Soc. 123 (1995), 1527-1535 Request permission

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
  • © Copyright 1995 American Mathematical Society
  • 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