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Zbl 0947.11011
Hauss, Michael
A Boole-type formula involving conjugate Euler polynomials. (English)
[A] Butzer, P. L. (ed.) et al., Karl der Grosse und sein Nachwirken. 1200 Jahre Kultur und Wissenschaft in Europa. Band 2: Mathematisches Wissen. Turnhout: Brepols. 361-375 (1998). ISBN 2-503-50674-7/hbk

The conjugate Euler polynomials $E^\sim_n(x)$ are defined by the application of the Hilbert transform to the (periodic) Euler polynomials ${\cal E}_n(x)$. In the paper under review an analogue of the Boole summation formula is proved, where the Euler polynomials are replaced by the conjugate ones. As an application, partial fraction expansions are given from which an Euler--type formula follows, namely $$ \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{2m}}=(-1)^m \pi^{2m} \frac{E^\sim_{2m-1}(\frac 12)}{(2m-1)!} $$ for $m\in{\Bbb N}$, analogue to the case of an odd exponent $2m-1$. But in contrast to the odd case $2m-1$, it remains as an open question whether the factor of $\pi^{2m}$ is rational or not.
[Helmut Müller (Hamburg)]

MSC 2000:: *11B68 Bernoulli numbers, etc.

Keywords: Euler polynomials

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