Difference between revisions of "Euler E"

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__NOTOC__
 
The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by
 
The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by
$$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!}.$$
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$$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$
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where $e_k$ denotes an [[Euler number]].
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*$E_0(x)=1$
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*$E_1(x)=x-\dfrac{1}{2}$
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*$E_2(x)=x^2-x$
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*$E_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}$
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*$E_4(x)=x^4-2x^3+x$
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=Properties=
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[[Euler E generating function]]<br />
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[[Euler E n'(x)=nE n-1(x)]]<br />
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<div class="toccolours mw-collapsible mw-collapsed">
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<strong>Theorem:</strong> The following formula holds:
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$$E_n(x+y)=\displaystyle\sum_{k=0}^n {n \choose k} E_k(x)y^k.$$
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<div class="mw-collapsible-content">
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<strong>Proof:</strong> █
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</div>
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</div>
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{{:Orthogonal polynomials footer}}
 
{{:Orthogonal polynomials footer}}
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[[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:


Orthogonal polynomials