調和数・一般化調和数

\[H_{n}^{\left(1\right)}=H_{n}\]

\[H_{n}^{\left(r\right)}:=\begin{cases}\frac{1}{n} & r=0\\\sum_{k=1}^{n}H_{k}^{\left(r-1\right)} & r\in\mathbb{N}\end{cases}\]

\[H_{1-z}-H_{z}=\pi\tan^{-1}\left(\pi z\right)+\frac{1}{1-z}-\frac{1}{z}\]

\[H_{z}=\sum_{k=1}^{\infty}\left(-1\right)^{k-1}\frac{1}{k}C\left(z,k\right)\]

\[H_{nz,m}=\frac{n^{m-1}-1}{n^{m-1}}\zeta\left(m\right)+\frac{1}{n^{m}}\sum_{k=0}^{n-1}H_{z+\frac{k}{n},m}\]

\[\frac{d^{n}H_{z,\alpha}}{dz^{n}}=\zeta\left(\alpha\right)\delta_{0n}+\left(-1\right)^{n+1}Q\left(\alpha,n\right)\left(\zeta\left(\alpha+n\right)-H_{z,\alpha+n}\right)\]

\[\int H_{z}dz=\log\Gamma\left(z+1\right)+\gamma z+C\]

\[H_{\frac{1}{2},z}=2^{z}-\left(2^{z}-2\right)\zeta\left(z\right)\]

\[H_{\frac{1}{2}}=2-2\log2\]

\[H_{z,m}=\zeta\left(m\right)-\zeta\left(m,z+1\right)\]

\[\sum_{k=1}^{n}H_{k,m}=\left(n+1\right)H_{n,m}-H_{n,m-1}\]

\[H_{n,m}:=\sum_{k=1}^{n}\frac{1}{k^{m}}\]