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A Boole-type formula involving conjugate Euler polynomials. (English) Zbl 0947.11011
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).

The conjugate Euler polynomials E n (x) are defined by the application of the Hilbert transform to the (periodic) Euler polynomials 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

k=0 (-1) k (2k+1) 2m =(-1) m π 2m E 2m-1 (1 2) (2m-1)!

for m, 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 π 2m is rational or not.

11B68Bernoulli and Euler numbers and polynomials