Difference between revisions of "Bernoulli B"

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(Created page with "Bernoulli polynomials $B_n$ are defined by the formula $$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} \dfrac{B_n(x) t^n}{n!}.$$")
 
 
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Bernoulli polynomials $B_n$ are defined by the formula
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Bernoulli polynomials $B_n$ are [[orthogonal polynomials]] defined by the formula
$$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} \dfrac{B_n(x) t^n}{n!}.$$
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$$B_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} b_{n-k}x^k,$$
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where $b_k$ are [[Bernoulli number|Bernoulli numbers]].
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$$B_0(x)=1$$
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$$B_1(x)=x-\dfrac{1}{2}$$
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$$B_2(x)=x^2-x+\dfrac{1}{6}$$
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$$B_3(x)=x^3-\dfrac{3x^2}{2}+\dfrac{x}{2}$$
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$$B_4(x)=x^4-2x^3+x^2-\dfrac{1}{30}$$
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=Properties=
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[[Bernoulli polynomial and Hurwitz zeta]]<br />
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=See Also=
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[[Bernoulli numbers]]<br />
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{{: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