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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to $\zeta(2m+1)$. (English) Zbl 0877.11009
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 $B^{\sim}_{2m+1}=B_{2m+1}^{\sim}(0)$.

11B68Bernoulli and Euler numbers and polynomials