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Bernoulli numbers and polynomials

Table of contents: Tables - Generating functions - Sum representations - Representations by special functions - Specific values - Functional equations - Derivatives and integrals - Summation - Denominators - Bounds and inequalities

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Symbol: BernoulliB BnB_{n} Bernoulli number
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Symbol: BernoulliPolynomial Bn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial

Tables

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Table of BnB_{n} for 0n500 \le n \le 50
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Table of Bn ⁣(x)B_{n}\!\left(x\right) for 0n100 \le n \le 10

Generating functions

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zez1=n=0Bnznn!\frac{z}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n} \frac{{z}^{n}}{n !}
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zexzez1=n=0Bn ⁣(x)znn!\frac{z {e}^{x z}}{{e}^{z} - 1} = \sum_{n=0}^{\infty} B_{n}\!\left(x\right) \frac{{z}^{n}}{n !}

Sum representations

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Bn ⁣(x)=k=0n(nk)BnkxkB_{n}\!\left(x\right) = \sum_{k=0}^{n} {n \choose k} B_{n - k} {x}^{k}

Representations by special functions

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B2n=(1)n+12(2n)!ζ ⁣(2n)(2π)2nB_{2 n} = {\left(-1\right)}^{n + 1} \frac{2 \left(2 n\right)! \zeta\!\left(2 n\right)}{{\left(2 \pi\right)}^{2 n}}

Specific values

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B2n+3=0B_{2 n + 3} = 0
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(1)n+1B2n+2>0{\left(-1\right)}^{n + 1} B_{2 n + 2} \gt 0
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Bn ⁣(0)=BnB_{n}\!\left(0\right) = B_{n}
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Bn ⁣(1)=(1)nBnB_{n}\!\left(1\right) = {\left(-1\right)}^{n} B_{n}
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Bn ⁣(12)=(21n1)BnB_{n}\!\left(\frac{1}{2}\right) = \left({2}^{1 - n} - 1\right) B_{n}

Functional equations

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Bn ⁣(x+1)=Bn ⁣(z)+nxn1B_{n}\!\left(x + 1\right) = B_{n}\!\left(z\right) + n {x}^{n - 1}
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Bn ⁣(1x)=(1)nBn ⁣(x)B_{n}\!\left(1 - x\right) = {\left(-1\right)}^{n} B_{n}\!\left(x\right)
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Bn ⁣(x)=(1)n(Bn ⁣(x)+nxn1)B_{n}\!\left(-x\right) = {\left(-1\right)}^{n} \left(B_{n}\!\left(x\right) + n {x}^{n - 1}\right)

Derivatives and integrals

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abBn ⁣(t)dt=Bn+1 ⁣(b)Bn+1 ⁣(a)n+1\int_{a}^{b} B_{n}\!\left(t\right) \, dt = \frac{B_{n + 1}\!\left(b\right) - B_{n + 1}\!\left(a\right)}{n + 1}
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zz+1Bn ⁣(t)dt=zn\int_{z}^{z + 1} B_{n}\!\left(t\right) \, dt = {z}^{n}
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ddxBn ⁣(x)=nBn1 ⁣(x)\frac{d}{d x}\, B_{n}\!\left(x\right) = n B_{n - 1}\!\left(x\right)

Summation

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k=1mkn=Bn+1 ⁣(m+1)Bm+1m+1\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}

Denominators

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(B2n+(p1)2n1p)Z\left(B_{2 n} + \sum_{\left(p - 1\right) \mid 2 n} \frac{1}{p}\right) \in \mathbb{Z}
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(B2n(p1)2n1p)Z\left(B_{2 n} \prod_{\left(p - 1\right) \mid 2 n} \frac{1}{p}\right) \in \mathbb{Z}

Bounds and inequalities

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B2n<(1+1n)2(2n)!(2π)2n\left|B_{2 n}\right| \lt \left(1 + \frac{1}{n}\right) \frac{2 \left(2 n\right)!}{{\left(2 \pi\right)}^{2 n}}
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B2n<(1+1n)4πn(nπe)2n\left|B_{2 n}\right| \lt \left(1 + \frac{1}{n}\right) 4 \sqrt{\pi n} {\left(\frac{n}{\pi e}\right)}^{2 n}
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B2n>2(2n)!(2π)2n\left|B_{2 n}\right| \gt \frac{2 \left(2 n\right)!}{{\left(2 \pi\right)}^{2 n}}
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B2n>4πn(nπe)2n\left|B_{2 n}\right| \gt 4 \sqrt{\pi n} {\left(\frac{n}{\pi e}\right)}^{2 n}
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B2n ⁣(x)B2n\left|B_{2 n}\!\left(x\right)\right| \le \left|B_{2 n}\right|
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B2n ⁣(x)<B2n\left|B_{2 n}\!\left(x\right)\right| \lt \left|B_{2 n}\right|

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2019-07-15 23:42:41.550119 UTC