巾関数と多重対数関数の積の積分

巾関数と多重対数関数の積の積分

\(n\in\mathbb{Z}\)とする。

\[ \int z^{\alpha}\Li_{n}\left(z\right)dz=\frac{\left(-1\right)^{n}z^{\alpha+1}}{\left(\alpha+1\right)^{n}}\left\{ \sum_{k=1}^{n}\left(\left(-1\right)^{k}\left(\alpha+1\right)^{k-1}\Li_{k}\left(z\right)\right)+\frac{z}{\alpha+2}F\left(1,\alpha+2;\alpha+3;z\right)\right\} \]

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\(\Li_{n}\left(x\right)\)は多重対数関数

\begin{align*} \int z^{\alpha}\Li_{n}\left(z\right)dz & =\frac{1}{\alpha+1}z^{\alpha+1}\Li_{n}\left(z\right)-\frac{1}{\alpha+1}\int z^{\alpha}\Li_{n-1}\left(z\right)dz\\ & =\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}\left\{ \sum_{k=1}^{n}\left\{ \left(-1\right)^{k}\left(\alpha+1\right)^{k}\int z^{\alpha}\Li_{k}\left(z\right)dz-\left(-1\right)^{k-1}\left(\alpha+1\right)^{k-1}\int z^{\alpha}\Li_{k-1}\left(z\right)dz\right\} +\int z^{\alpha}\Li_{0}\left(z\right)dz\right\} \\ & =\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}\left\{ \sum_{k=1}^{n}\left(\left(-1\right)^{k}\left(\alpha+1\right)^{k-1}z^{\alpha+1}\Li_{k}\left(z\right)\right)+\int z^{\alpha+1}F\left(1;;z\right)dz\right\} \\ & =\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}\left\{ z^{\alpha+1}\sum_{k=1}^{n}\left(\left(-1\right)^{k}\left(\alpha+1\right)^{k-1}\Li_{k}\left(z\right)\right)+\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\alpha+2;\alpha+3;z\right)\right\} \\ & =\frac{\left(-1\right)^{n}z^{\alpha+1}}{\left(\alpha+1\right)^{n}}\left\{ \sum_{k=1}^{n}\left(\left(-1\right)^{k}\left(\alpha+1\right)^{k-1}\Li_{k}\left(z\right)\right)+\frac{z}{\alpha+2}F\left(1,\alpha+2;\alpha+3;z\right)\right\} \end{align*}

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巾関数と多重対数関数の積の積分

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