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An Euler-Maclaurin-type formula involving conjugate Bernoulli polynomials and an application to ζ(2m+1). (English) Zbl 0877.11009

The conjugate Bernoulli polynomials B n (x) mentioned in the title are defined by applying the Hilbert transform to the (1-periodic) Bernoulli polynomials, B n (x)=H 1 n (x), x[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 n (x) are replaced by B n (x). As an application the partial fraction expansion of the generating function of the B n (x) is given, from which the remarkable Euler formula for ζ(2m+1),

ζ(2m+1)=(-1) m 2 2m π 2m+1 B 2m+1 (2m+1)!,m

can be deduced with the conjugate Bernoulli number B 2m+1 =B 2m+1 (0).

MSC:
11B68Bernoulli and Euler numbers and polynomials