Euler polynomial


The Euler polynomialsE0(x),E1(x),E2(x),  are certain polynomialsPlanetmathPlanetmath of the indeterminate x with rational coefficients (whose denominators may only be powers 1, 2, 4, 8,  of 2).  The Euler polynomials may be defined by means of the generating function such that

2extet+1=n=0En(x)tnn!,

i.e. one can get them by dividing the Taylor series 2+2xt+x2t2+13x3t3+  by the Taylor series 2+t+12t2+16t3+.  There are also explicit formulae for the polynomials, e.g.

En(x)=k=0n(nk)Ek2k(x-12)n-k

via the Euler numbersMathworldPlanetmath Ek.  Conversely, the Euler numbers are expressed with the Euler polynomials through

Ek= 2kEk(12).

The first seven Euler polynomials are

E0(x)= 1
E1(x)=x-12
E2(x)=x2-x
E3(x)=x3-32x2+14
E4(x)=x4-2x3+x
E5(x)=x5-52x4+52x2-12
E6(x)=x6-3x5+5x3-3x

The Euler polynomials have the beautiful addition formulaPlanetmathPlanetmath

En(x+y)=k=0n(nk)Ek(x)yk

and the derivativePlanetmathPlanetmath

En(x)=nEn-1(x)  (for n=1, 2,).

The Euler polynomials form an example of Appell sequences.

Title Euler polynomial
Canonical name EulerPolynomial
Date of creation 2013-03-22 19:07:07
Last modified on 2013-03-22 19:07:07
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition
Classification msc 11B68
Related topic BernoulliPolynomial
Related topic BernoulliPolynomialsAndNumbers