# Euler polynomial

The Euler polynomials$E_{0}(x),\,E_{1}(x),\,E_{2}(x),\,\ldots$  are certain polynomials of the indeterminate $x$ with rational coefficients (whose denominators may only be powers $1,\,2,\,4,\,8,\,\ldots$  of 2).  The Euler polynomials may be defined by means of the generating function such that

 $\frac{2e^{xt}}{e^{t}\!+\!1}\;=\;\sum_{n=0}^{\infty}E_{n}(x)\frac{t^{n}}{n!},$

i.e. one can get them by dividing the Taylor series $2+2xt+x^{2}t^{2}+\frac{1}{3}x^{3}t^{3}+\ldots$  by the Taylor series $2+t+\frac{1}{2}t^{2}+\frac{1}{6}t^{3}+\ldots$.  There are also explicit formulae for the polynomials, e.g.

 $E_{n}(x)\;=\;\sum_{k=0}^{n}{n\choose k}\frac{E_{k}}{2^{k}}\left(x\!-\!\frac{1}% {2}\right)^{n-k}$

via the Euler numbers $E_{k}$.  Conversely, the Euler numbers are expressed with the Euler polynomials through

 $E_{k}\;=\;2^{k}E_{k}\!\!\left(\!\frac{1}{2}\!\right).$

The first seven Euler polynomials are

 $\displaystyle E_{0}(x)\;=\;1$ $\displaystyle E_{1}(x)\;=\;x\!-\!\frac{1}{2}$ $\displaystyle E_{2}(x)\;=\;x^{2}\!-\!x$ $\displaystyle E_{3}(x)\;=\;x^{3}\!-\!\frac{3}{2}x^{2}\!+\!\frac{1}{4}$ $\displaystyle E_{4}(x)\;=\;x^{4}\!-\!2x^{3}\!+\!x$ $\displaystyle E_{5}(x)\;=\;x^{5}\!-\!\frac{5}{2}x^{4}\!+\!\frac{5}{2}x^{2}\!-% \!\frac{1}{2}$ $\displaystyle E_{6}(x)\;=\;x^{6}\!-\!3x^{5}\!+\!5x^{3}\!-\!3x$

The Euler polynomials have the beautiful addition formula

 $E_{n}(x\!+\!y)\;=\;\sum_{k=0}^{n}{n\choose k}E_{k}(x)y^{k}$

and the derivative

 $E_{n}^{\prime}(x)\;=\;nE_{n-1}(x)\qquad(\textrm{for }n=1,\,2,\,\ldots).$

The Euler polynomials form an example of Appell sequences.

Title Euler polynomial EulerPolynomial 2013-03-22 19:07:07 2013-03-22 19:07:07 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 11B68 BernoulliPolynomial BernoulliPolynomialsAndNumbers