TITLE:
Obtaining Simply Explicit Form and New Properties of Euler Polynomials by Differential Calculus
AUTHORS:
Do Tan Si
KEYWORDS:
Obtaining Appell Type Euler Numbers and Polynomials, Relations Euler-Bernoulli Polynomials, Sums over km, Series on k-m, Euler Series of Functions
JOURNAL NAME:
Applied Mathematics,
Vol.14 No.7,
July
27,
2023
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 πm as series on k-m and for expanding functions into series of Euler polynomials.