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Zbl 0877.11009
Hauss, Michael
An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta(2m+1)$.
(English)
[J] Commun. Appl. Anal. 1, No.1, 15-32 (1997). ISSN 1083-2564

The conjugate Bernoulli polynomials $B_n^{\sim}(x)$ mentioned in the title are defined by applying the Hilbert transform to the ($1$-periodic) Bernoulli polynomials, $B_n^{\sim}(x)= H_1{\cal B}_n(x)$, $x\in[0,1)$. A generating function as well as several representations of the conjugate Bernoulli polynomials are given; furthermore an analogue to the famous Euler-Maclaurin summation formula is obtained, where the classical Bernoulli polynomials ${\cal B}_n(x)$ are replaced by $B_n^{\sim}(x)$. As an application the partial fraction expansion of the generating function of the $B_n^{\sim}(x)$ is given, from which the remarkable Euler formula for $\zeta(2m+1)$, $$\zeta(2m+1)= (-1)^m 2^{2m} \pi^{2m+1}\frac{B^{\sim}_{2m+1}}{(2m+1)!},\quad m\in\Bbb N$$ can be deduced with the conjugate Bernoulli number $B^{\sim}_{2m+1}=B_{2m+1}^{\sim}(0)$.
[H.Müller (Hamburg)]
MSC 2000:
*11B68 Bernoulli numbers, etc.

Keywords: generalized Bernoulli polynomials; generating function; conjugate Bernoulli polynomials; Euler-Maclaurin summation formula; conjugate Bernoulli number

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