Difference between revisions of "Euler E"
From specialfunctionswiki
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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 | ||
$$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ | $$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ | ||
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=Properties= | =Properties= | ||
| − | + | [[Euler E generating function]]<br /> | |
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Revision as of 17:45, 24 June 2016
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: █
