Euler polynomial
The Euler polynomials E0(x),E1(x),E2(x),… are certain polynomials 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)=n∑k=0(nk)Ek2k(x-12)n-k |
via the Euler numbers 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 formula
En(x+y)=n∑k=0(nk)Ek(x)yk |
and the derivative
E′n(x)=nEn-1(x) |
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 |