Difference between revisions of "Euler E"

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The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by
 
The Euler polynomials $E_n(x)$ are [[orthogonal polynomials]] defined by
$$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!}.$$
+
$$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$
 +
where $e_k$ denotes an [[Euler number]].
 +
 
 +
*$E_0(x)=1$
 +
*$E_1(x)=x-\dfrac{1}{2}$
 +
*$E_2(x)=x^2-x$
 +
*$E_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}$
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*$E_4(x)=x^4-2x^3+x$
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 +
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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<strong>Theorem:</strong> The following formula holds:
 +
$$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!},$$
 +
where $e^{xt}$ denotes the [[exponential function]] and $E_n$ denotes an [[Euler E]] polynomial.
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<div class="mw-collapsible-content">
 +
<strong>Proof:</strong>
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</div>
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</div>
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{{:Orthogonal polynomials footer}}
 
{{:Orthogonal polynomials footer}}

Revision as of 10:55, 23 March 2015

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: $$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!},$$ where $e^{xt}$ denotes the exponential function and $E_n$ denotes an Euler E polynomial.

Proof:


Orthogonal polynomials