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 | + | 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,$$ |
| + | where $b_k$ are [[Bernoulli number|Bernoulli numbers]]. | ||
| + | |||
| + | $$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}$$ | ||
| + | |||
| + | =Properties= | ||
| + | [[Bernoulli polynomial and Hurwitz zeta]]<br /> | ||
| + | |||
| + | =See Also= | ||
| + | [[Bernoulli numbers]]<br /> | ||
| + | {{:Orthogonal polynomials footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 22:46, 20 June 2016
Bernoulli polynomials $B_n$ are orthogonal polynomials defined by the formula $$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
