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Thread: Bernoulli Polynomial

  1. #1
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    Bernoulli Polynomial

    By considering $\displaystyle \overset{1}{\underset{0}{\int}}xB_{n}(x)\, dx$.

    I am trying to prove that

    $\displaystyle B_{n+1}=(n+1)\overset{n}{\underset{r=0}{\sum}}\lef t(\begin{array}{c}
    n\\
    r\end{array}\right)\frac{B_{r}}{n-r+2}$


    but failing miserably, despite trying to integrate by parts.

    Any takers?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Cairo View Post
    By considering $\displaystyle \overset{1}{\underset{0}{\int}}xB_{n}(x)\, dx$.

    I am trying to prove that

    $\displaystyle B_{n+1}=(n+1)\overset{n}{\underset{r=0}{\sum}}\lef t(\begin{array}{c}
    n\\
    r\end{array}\right)\frac{B_{r}}{n-r+2}$


    but failing miserably, despite trying to integrate by parts.

    Any takers?
    What definition are you taking as the Bernoulli polynomials?
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  3. #3
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    Sorry. I should have said that

    $\displaystyle \frac{ze^{xz}}{e^{z}-1}=\overset{\infty}{\underset{n=0}{\sum}}\frac{B_{ n}(x)}{n!}z^{n}$

    where

    $\displaystyle B_{n}(x)=\overset{\infty}{\underset{k=0}{\sum}}\le ft(\begin{array}{c}
    n\\
    k\end{array}\right)B_{k}x^{n-k}$.
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