Euler E

The Euler polynomials $E_n(x)$ are orthogonal polynomials defined by $$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ where $e_k$ denotes an Euler number.

• $E_0(x)=1$
• $E_1(x)=x-\dfrac{1}{2}$
• $E_2(x)=x^2-x$
• $E_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}$
• $E_4(x)=x^4-2x^3+x$

Properties

Theorem: The following formula holds: $$E_n(x+y)=\displaystyle\sum_{k=0}^n {n \choose k} E_k(x)y^k.$$

Proof:

Theorem: The following formula holds: $$E_n'(x)=nE_{n-1}(x);n=1,2,\ldots.$$

Proof:

Orthogonal polynomials