Euler polynomial

The Euler polynomials  $E_0(x),E_1(x),E_2(x),\mathrm{\dots }$  are certain polynomials of the indeterminate $x$ with rational coefficients (whose denominators may only be powers $1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}8},\mathrm{\dots }$  of 2).  The Euler polynomials may be defined by means of the generating function such that

$$\frac{2e^{xt}}{e^t+1}=\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!},$$

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

$$E_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{E_k}{2^k}{\left(x-\frac{1}{2}\right)}^{n-k}$$

via the Euler numbers $E_k$.  Conversely, the Euler numbers are expressed with the Euler polynomials through

$$E_k={\mathrm{\hspace{0.33em}2}}^k{E}_k\left(\frac{1}{2}\right).$$

The first seven Euler polynomials are

$E_0(x)=\mathrm{\hspace{0.33em}1}$

$E_1(x)=x-{\displaystyle \frac{1}{2}}$

$E_2(x)=x^2-x$

$E_3(x)=x^3-{\displaystyle \frac{3}{2}}{x}^2+{\displaystyle \frac{1}{4}}$

$E_4(x)=x^4-2x^3+x$

$E_5(x)=x^5-{\displaystyle \frac{5}{2}}{x}^4+{\displaystyle \frac{5}{2}}{x}^2-{\displaystyle \frac{1}{2}}$

$E_6(x)=x^6-3x^5+5x^3-3x$

The Euler polynomials have the beautiful addition formula

$$E_n(x+y)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k(x)y^k$$

and the derivative

$$E_n^{\prime }(x)=n{E}_{n-1}(x)\mathit{\hspace{1em}\hspace{1em}}(\text{for }n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots }).$$

The Euler polynomials form an example of Appell sequences.