Eulerian polynomials
نویسندگان
چکیده
A short report on my lecture at the Conference in Honor of W. Killing in Münster on December 7, 2007, together with some remarks on the preceding paper by ArjehM. Cohen. I reported on Euler’s famous paper Remarques sur un beau rapport entre les series des puissances tant direct que reciproques, Académie des sciences de Berlin, Lu en 1749, Opera Omnia Serie I, Bd. 15, pp. 70-90. In modern terminology Euler introduces the alternating ζ-function φ(s) = 1− 1 2s + 1 3s − 1 4s + . . . . This series converges for Re(s) > 0. The function φ(s) can be holomorphically extended to the whole s-plane. It is related to the ζ-function: φ(s) = (1 − 2)ζ(s). Euler wants to calculate φ(−n) for n = 0, 1, 2, 3, . . . . For this purpose he introduces the Eulerian polynomials Pn(t) by (1) ∞
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