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Some sum relations involving Bernoulli and Euler polynomials. (English) Zbl 1217.11021
The classical Bernoulli polynomials and Euler polynomials are defined by means of the following generating functions: \[ \frac{ze^{xz}}{e^{z}-1}=\sum_{n=0}^{\infty }B_{n}(x) \frac{z^{n}}{n!}\quad ( \left| z\right| <2\pi) \] and \[ \frac{2e^{xz}}{e^{z}+1}=\sum_{n=0}^{\infty }E_{n}(x) \frac{z^{n}}{n!}\quad ( \left | z \right | <\pi) \] respectively. Obviously, \(B_{n}:=B_{n}(0), \;E_{n}:=2^nE_{n}\left(\frac12\right)\) are the corresponding Bernoulli numbers and the Euler numbers, respectively.
In the present paper, the authors obtain several symmetric identities for the Bernoulli polynomials and Euler polynomials using the generating function method.

11B68 Bernoulli and Euler numbers and polynomials
39A70 Difference operators
Full Text: DOI
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