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| | =Properties= | | =Properties= |
| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} B_k(x)\dfrac{t^k}{k!}.$$
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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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| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$\displaystyle\int_a^x B_n(t) dt = \dfrac{B_{n+1}(x)-B_{n+1}(a)}{n+1}.$$
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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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| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$B_n(mx)=m^{n-1}\displaystyle\sum_{k=0}^{m-1} B_n \left( x + \dfrac{k}{m} \right).$$
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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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| | [[Bernoulli polynomial and Hurwitz zeta]]<br /> | | [[Bernoulli polynomial and Hurwitz zeta]]<br /> |
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Latest revision as of 22:46, 20 June 2016
Bernoulli polynomials $B_n$ are orthogonal polynomials defined by the formula
$$B_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} b_{n-k}x^k,$$
where $b_k$ are Bernoulli numbers.
$$B_0(x)=1$$
$$B_1(x)=x-\dfrac{1}{2}$$
$$B_2(x)=x^2-x+\dfrac{1}{6}$$
$$B_3(x)=x^3-\dfrac{3x^2}{2}+\dfrac{x}{2}$$
$$B_4(x)=x^4-2x^3+x^2-\dfrac{1}{30}$$
Properties
Bernoulli polynomial and Hurwitz zeta
See Also
Bernoulli numbers
Orthogonal polynomials
Bernoulli $B$
Bernoulli $B$
