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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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.

References [Enhancements On Off] (What's this?)

  • [1] Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun, Dover Publications, Inc., New York, 1966. MR 0208797 (34 #8606)
  • [2] A. Erdeley, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, Vol. 1, McGraw-Hill, New York, 1953.
  • [3] A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An index of mathematical tables (2nd ed.), Blackwell Scientific Publications, Oxford, England, 1962.
  • [4] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York, 1980.
  • [5] E. R. Hansen, A table of series and products, Prentice-Hall, Englewood Cliffs, NJ, 1975.
  • [6] C. Jordan, Calculus of finite differences, Chelsea, New York, 1947.
  • [7] Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968 (38 #1291)
  • [8] L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan and Co., Ltd., London, 1951. MR 0043339 (13,245c)
  • [9] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.
  • [10] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990. More special functions; Translated from the Russian by G. G. Gould. MR 1054647 (91c:33001)
  • [11] E. L. Stark, ∑^{∞}_{𝑘=1}𝐾^{-𝑠}, 𝑠=2,3,4\cdots, once more, Math. Mag. 47 (1974), 197–202. MR 0352775 (50 #5261)

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Additional Information

PII: S 0002-9939(1995)1283544-0
Keywords: Bernoulli polynomials, Euler polynomials, Hurwitz zeta function, summation of series
Article copyright: © Copyright 1995 American Mathematical Society