Difference between revisions of "Euler E"
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=Properties= | =Properties= | ||
{{:Euler E generating function}} | {{:Euler E generating function}} | ||
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| + | <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> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$E_n'(x)=nE_{n-1}(x);n=1,2,\ldots.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
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{{:Orthogonal polynomials footer}} | {{:Orthogonal polynomials footer}} | ||
Revision as of 10:58, 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.
- $E_0(x)=1$
- $E_1(x)=x-\dfrac{1}{2}$
- $E_2(x)=x^2-x$
- $E_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}$
- $E_4(x)=x^4-2x^3+x$
Contents
Properties
Theorem
The following formula holds for $|z|<\pi$: $$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!},$$ where $e^{xt}$ denotes the exponential function and $E_n$ denotes an Euler E polynomial.
Proof
References
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.14 (2)$
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: █
