Difference between revisions of "Euler E"
From specialfunctionswiki
(→Properties) |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| + | __NOTOC__ | ||
The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by | The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by | ||
$$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ | $$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ | ||
| Line 10: | Line 11: | ||
=Properties= | =Properties= | ||
| − | + | [[Euler E generating function]]<br /> | |
| + | [[Euler E n'(x)=nE n-1(x)]]<br /> | ||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
<strong>Theorem:</strong> The following formula holds: | <strong>Theorem:</strong> The following formula holds: | ||
$$E_n(x+y)=\displaystyle\sum_{k=0}^n {n \choose k} E_k(x)y^k.$$ | $$E_n(x+y)=\displaystyle\sum_{k=0}^n {n \choose k} E_k(x)y^k.$$ | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Latest revision as of 01:05, 4 March 2018
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
Euler E generating function
Euler E n'(x)=nE n-1(x)
Theorem: The following formula holds: $$E_n(x+y)=\displaystyle\sum_{k=0}^n {n \choose k} E_k(x)y^k.$$
Proof: █
