ファウルハーバーの公式

この記事では, ファウルハーバーの公式という, $m$乗和の明示式について, 軽く証明を書こうと思います.

ベルヌーイ数$B_n$を ${\displaystyle \frac{x}{e^x-1}=\sum _{n=0}^{\mathrm{\infty }}\frac{B_n}{n!}{x}^n}$ で定めます.

$$\begin{array}{rcl}\sum _{k=1}^n{k}^m& =& \sum _{k=1}^n{\left(\frac{d}{dx}\right)}^m{e}^{kx}{|}_{x=0}\\ & =& {\left(\frac{d}{dx}\right)}^m\frac{e^{(n+1)x}-e^x}{e^x-1}{|}_{x=0}\\ & =& {\left(\frac{d}{dx}\right)}^m\frac{x{e}^x}{e^x-1}\cdot \frac{e^{nx}-1}{x}{|}_{x=0}\end{array}$$

ここで,

$$\begin{array}{rcl}{\left(\frac{d}{dx}\right)}^k\frac{x{e}^x}{e^x-1}{|}_{x=0}& =& {\left(\frac{d}{dx}\right)}^k\frac{-x}{e^{-x}-1}{|}_{x=0}\\ & =& (-1{)}^k{B}_k\\ {\left(\frac{d}{dx}\right)}^k\frac{e^{nx}-1}{x}{|}_{x=0}& =& {\left(\frac{d}{dx}\right)}^k\sum _{j=1}^{\mathrm{\infty }}\frac{n^j}{j!}{x}^{j-1}{|}_{x=0}\\ & =& \frac{n^{k+1}}{k+1}\end{array}$$

と, 積の微分公式 ${\displaystyle {\left(\frac{d}{dx}\right)}^{m}f(x)g(x)=\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k})f^{(k)}(x)g^{(m-k)}(x)}$ より,
$$\begin{array}{rcl}\sum _{k=1}^n{k}^m& =& {\left(\frac{d}{dx}\right)}^m\frac{x{e}^x}{e^x-1}\cdot \frac{e^{nx}-1}{x}{|}_{x=0}\\ & =& \sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k})(-1{)}^{m-k}{B}_{m-k}\frac{n^{k+1}}{k+1}\\ & =& \frac{1}{m+1}\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m+1}{k+1})(-1{)}^{m-k}{B}_{m-k}{n}^{k+1}\\ & =& \frac{1}{m+1}\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m+1}{k})(-1{)}^k{B}_k{n}^{m-k+1}\end{array}$$

示すことができました.
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