New formulae for the Bernoulli and Euler polynomials at rational arguments. (English) Zbl 0827.11012

Cvijović, Djurdje; Klinowski, Jacek

Proc. Am. Math. Soc. 123, No. 5, 1527-1535 (1995).

Authors’ summary: We prove theorems on the values of the Bernoulli polynomials \(B_n (x)\) with \(n=2, 3,\dots\) and the Euler polynomials \(E_n (x)\) with \(n=2, 3,\dots\) for \(0<x <1\) where \(x\) is rational. \(B_n (x)\) and \(E_n (x)\) are expressible in terms of a finite combination of trigonometric functions and the Hurwitz zeta function \(\zeta (z,a)\). The well known argument addition formulae and reflection property of \(B_n (x)\) and \(E_n (x)\) extend this result to any rational argument. We also deduce new relations concerning the finite sums of the Hurwitz zeta function and sum some classical trigonometric series.

Reviewer: D.Kershaw (Lancaster)

MSC:

11B68	Bernoulli and Euler numbers and polynomials
33E99	Other special functions

Keywords:

Bernoulli polynomials; Euler polynomials; Hurwitz zeta function; trigonometric series