Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus

Abstract

Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums; may be all its relations with Bernoulli polynomials, Bernoulli numbers; its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained; the formulae for obtaining all π m as series on k -m and for expanding functions into series of Euler polynomials.