| Version 3 |
Version 2 |
| For\, $n = 0,\,1,\,2,\,\ldots$,\, the {\em Bernoulli polynomial} may be defined as the uniquely determined polynomial $b_n(x)$ satisfying |
For\, $n = 0,\,1,\,2,\,\ldots$,\, the {\em Bernoulli polynomial} may be defined as the uniquely determined polynomial $b_n(x)$ satisfying |
| \begin{align} |
\begin{align} |
| \int_x^{x+1}b_n(t)\,dt = x^n. |
\int_x^{x+1}b_n(t)\,dt = x^n. |
| \end{align} |
\end{align} |
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| The constant term of $b_n(x)$ is the $n^{\mathrm{th}}$ {\em Bernoulli number} $B_n$. |
The constant term of $b_n(x)$ is the $n^{\mathrm{th}}$ {\em Bernoulli number} $B_n$. |
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| The Bernoulli polynomial is often denoted also $B_n(x)$.\\ |
The Bernoulli polynomial is often denoted also $B_n(x)$.\\ |
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| The uniqueness of the solution $b_n(x)$ in (1) is justificated by the |
The uniqueness of the solution $b_n(x)$ in (1) is justificated by the |
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| \textbf{Lemma.}\, For any polynomial $f(x)$, there exists a unique polynomial $g(x)$ with the same degree satisfying |
\textbf{Lemma.}\, For any polynomial $f(x)$, there exists a unique polynomial $g(x)$ with the same degree satisfying |
| \begin{align} |
\begin{align} |
| \int_x^{x+1}g(t)\,dt = f(x). |
\int_x^{x+1}g(t)\,dt = f(x). |
| \end{align} |
\end{align} |
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| {\em Proof.}\, For every\, $n = 0,\,1,\,2,\,\ldots$,\, the polynomial |
{\em Proof.}\, For every\, $n = 0,\,1,\,2,\,\ldots$,\, the polynomial |
| $$g_n(x) := \int_x^{x+1}t^n\,dt = \frac{(x+1)^{n+1}-x^{n+1}}{n+1}$$ |
$$g_n(x) := \int_x^{x+1}t^n\,dt = \frac{(x+1)^{n+1}-x^{n+1}}{n+1}$$ |
| is monic and its degree is $n$.\, If the coefficient of $x^n$ in $f(x)$ is $a_0$, then the difference $f(x)\!-\!a_0g_n(x)$ is a polynomial of degree $n\!-\!1$.\, Correspondingly we obtain $f(x)-a_0g_n(x)-a_1g_{n-1}(x)$ having the degree $n\!-\!2$ and so on.\, Finally we see that |
is monic and its degree is $n$.\, If the coefficient of $x^n$ in $f(x)$ is $a_0$, then the difference $f(x)\!-\!a_0g_n(x)$ is a polynomial of degree $n\!-\!1$.\, Correspondingly we obtain $f(x)-a_0g_n(x)-a_1g_{n-1}(x)$ having the degree $n\!-\!2$ and so on.\, Finally we see that |
| $$f(x)-a_0g_n(x)-a_1g_{n-1}(x)-\ldots-a_ng_0(x)$$ |
$$f(x)-a_0g_n(x)-a_1g_{n-1}(x)-\ldots-a_ng_0(x)$$ |
| must be the zero polynomial.\, Therefore |
must be the zero polynomial.\, Therefore |
| \begin{align*} |
\begin{align*} |
| f(x) & = a_0g_n(x)+a_1g_{n-1}(x)+\ldots+a_ng_0(x)\\ |
f(x) & = a_0g_n(x)+a_1g_{n-1}(x)+\ldots+a_ng_0(x)\\ |
| & = \sum_{i=0}^na_ig_{n-i}(x)\\ |
& = \sum_{i=0}^na_ig_{n-i}(x)\\ |
| & = \sum_{i=0}^na_i\int_x^{x+1}t^{n-i}\,dt\\ |
& = \sum_{i=0}^na_i\int_x^{x+1}t^{n-i}\,dt\\ |
| & = \int_x^{x+1}\sum_{i=0}^na_it^{n-i}\,dt |
& = \int_x^{x+1}\sum_{i=0}^na_it^{n-i}\,dt |
| \end{align*} |
\end{align*} |
| whence we have\, $\displaystyle g(x) = \sum_{i=0}^na_ix^{n-i}$.\\ |
whence we have\, $\displaystyle g(x) = \sum_{i=0}^na_ix^{n-i}$.\\ |
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| The proof implies also that the coefficients of $g(x)$ are rational, if the coefficients of $f(x)$ are such.\, So we know that all Bernoulli polynomials have only rational coefficients. |
The proof implies also that the coefficients of $g(x)$ are rational, if the coefficients of $f(x)$ are such.\, So we know that all Bernoulli polynomials have only rational coefficients. |
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| \begin{thebibliography}{7} |
\begin{thebibliography}{7} |
| \bibitem{MMP} \CYRM. \CYRM. \CYRP\cyro\cyrs\cyrt\cyrn\cyri\cyrk\cyro\cyrv: |
\bibitem{MMP} \CYRM. \CYRM. \CYRP\cyro\cyrs\cyrt\cyrn\cyri\cyrk\cyro\cyrv: |
| {\em \CYRV\cyrv\cyre\cyrd\cyre\cyrn\cyri\cyre\, \cyrv\, \cyrt\cyre\cyro\cyrr\cyri\cyryu\, \cyra\cyrl\cyrg\cyre\cyrb\cyrr\cyra\cyri\cyrch\cyre\cyrs\cyrk\cyri\cyrh \, |
{\em \CYRV\cyrv\cyre\cyrd\cyre\cyrn\cyri\cyre\, \cyrv\, \cyrt\cyre\cyro\cyrr\cyri\cyryu\, \cyra\cyrl\cyrg\cyre\cyrb\cyrr\cyra\cyri\cyrch\cyre\cyrs\cyrk\cyri\cyrh \, |
| \cyrch\cyri\cyrs\cyre\cyrl}. \,\CYRI\cyrz\cyrd\cyra\cyrt\cyre\cyrl\cyrsftsn\cyrs\cyrt\cyrv\cyro \, |
\cyrch\cyri\cyrs\cyre\cyrl}. \,\CYRI\cyrz\cyrd\cyra\cyrt\cyre\cyrl\cyrsftsn\cyrs\cyrt\cyrv\cyro \, |
| ``\CYRN\cyra\cyru\cyrk\cyra''. \CYRM\cyro\cyrs\cyrk\cyrv\cyra \,(1982). |
``\CYRN\cyra\cyru\cyrk\cyra''. \CYRM\cyro\cyrs\cyrk\cyrv\cyra \,(1982). |
| \end{thebibliography} |
\end{thebibliography} |
