Difference between revisions of "Bernoulli B"
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Bernoulli polynomials $B_n$ are [[orthogonal polynomials]] defined by the formula | 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)=\displaystyle\sum_{k=0}^n {n \choose k} b_{n-k}x^k,$$ |
| + | where $b_k$ are [[Bernoulli numbers]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} B_k(x)\dfrac{t^k}{k!}.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | {{:Bernoulli polynomial and Hurwitz zeta}} | ||
{{:Orthogonal polynomials footer}} | {{:Orthogonal polynomials footer}} | ||
Revision as of 09:57, 23 March 2015
Bernoulli polynomials $B_n$ are orthogonal polynomials defined by the formula $$B_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} b_{n-k}x^k,$$ where $b_k$ are Bernoulli numbers.
Contents
Properties
Theorem: The following formula holds: $$\dfrac{te^{xt}}{e^t-1} = \displaystyle\sum_{k=0}^{\infty} B_k(x)\dfrac{t^k}{k!}.$$
Proof: █
Theorem
The following formula holds: $$B_n(x)=-n \zeta(1-n,x),$$ where $B_n$ denotes the Bernoulli polynomial and $\zeta$ denotes the Hurwitz zeta function.
