Difference between revisions of "Euler E"
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| − | The Euler polynomials $E_n(x)$ are defined by | + | __NOTOC__ |
| − | $$\ | + | The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by |
| + | $$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ | ||
| + | where $e_k$ denotes an [[Euler number]]. | ||
| + | |||
| + | *$E_0(x)=1$ | ||
| + | *$E_1(x)=x-\dfrac{1}{2}$ | ||
| + | *$E_2(x)=x^2-x$ | ||
| + | *$E_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}$ | ||
| + | *$E_4(x)=x^4-2x^3+x$ | ||
| + | |||
| + | =Properties= | ||
| + | [[Euler E generating function]]<br /> | ||
| + | [[Euler E n'(x)=nE n-1(x)]]<br /> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$E_n(x+y)=\displaystyle\sum_{k=0}^n {n \choose k} E_k(x)y^k.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | |||
| + | {{:Orthogonal polynomials footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 01:05, 4 March 2018
The Euler polynomials $E_n(x)$ are orthogonal polynomials defined by $$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: █
