Difference between revisions of "Bernoulli B"

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=Properties=
 
=Properties=
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[[Bernoulli polynomial and Hurwitz zeta]]<br />
<strong>Theorem:</strong> The following formula holds:
 
$$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} B_k(x)\dfrac{t^k}{k!}.$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$\displaystyle\int_a^x B_n(t) dt = \dfrac{B_{n+1}(x)-B_{n+1}(a)}{n+1}.$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$B_n(mx)=m^{n-1}\displaystyle\sum_{k=0}^{m-1} B_n \left( x + \dfrac{k}{m} \right).$$
 
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<strong>Proof:</strong> █
 
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{{:Bernoulli polynomial and Hurwitz zeta}}
 
  
 
=See Also=
 
=See Also=
 
[[Bernoulli numbers]]<br />
 
[[Bernoulli numbers]]<br />
 
{{:Orthogonal polynomials footer}}
 
{{:Orthogonal polynomials footer}}
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[[Category:SpecialFunction]]

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