リーマン・ゼータ関数とディリクレ・イータ関数の導関数の特殊値
\[\zeta'\left(0\right)=-\Log\sqrt{2\pi}\]
多重対数関数を含む積分
\[\int\Li_{n}\left(z\right)dz=\sum_{k=0}^{n-2}\left\{ \left(-1\right)^{n-k}z\Li_{k+2}\left(z\right)\right\} -\left(-1\right)^{n}\left(z-\left(1-z\right)\Li_{1}\left(z\right)\right)+C\]
逆正接関数・逆双曲線正接関数と多重対数関数の関係
\[\Tan^{\bullet}z=\frac{i}{2}\left(-\Li_{1}\left(iz\right)+\Li_{1}\left(-iz\right)\right)\]
微分と積分の関係
\[f\left(x\right)=\int_{f^{\bullet}\left(a\right)}^{x}f'\left(x\right)dx-a\]
三角関数と双曲線関数の実部と虚部
\[\sin z=\sin\left(\Re\left(z\right)\right)\cosh\left(\Im\left(z\right)\right)+i\cos\left(\Re\left(z\right)\right)\sinh\left(\Im\left(z\right)\right)\]
2項係数の相加平均・相乗平均を含む極限
\[\lim_{n\rightarrow\infty}\sqrt[n]{\sqrt[n+1]{\prod_{k=0}^{n}C\left(n,k\right)}}=\sqrt{e}\]
三角関数と双曲線関数のn乗積分
\[\int\sin^{2n+m_{\pm}}xdx=\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}xdx\right\} \]
階乗と冪乗の極限
\[\lim_{n\rightarrow\infty}\frac{x^{n}}{n!}=0\]
(拡張)多重階乗の逆数和
\[\sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(\frac{\Gamma\left(n+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(n+\frac{b}{a}+1\right)}-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right)\]
(拡張)多重階乗と階乗の関係
\[\left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!}\]
整数と半整数の逆数和
\[\sum_{k=0}^{n}\frac{1}{k!}=e\frac{\Gamma(n+1,1)}{\Gamma\left(n+1\right)}\]
2重階乗の逆数和
\[\sum_{k=0}^{n}\frac{1}{\left(2k\right)!!}=\sqrt{e}\frac{\Gamma\left(n+1,\frac{1}{2}\right)}{\Gamma\left(n+1\right)}\]
第2種不完全ガンマ関数とガンマ関数の比の極限
\[\lim_{k\rightarrow0}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}=\delta_{0x}\]
第1種・第2種不完全ガンマ関数の基本性質
\[\Gamma\left(1,x\right)=e^{-x}\]
第1種・第2種不完全ガンマ関数の微分
\[\frac{\partial\Gamma\left(a,x\right)}{\partial x}=-x^{a-1}e^{-x}\]
第1種・第2種不完全ガンマ関数の漸化式
\[\Gamma\left(a+1,x\right)=a\Gamma\left(a,x\right)+x^{a}e^{-x}\]
不完全ガンマ関数とガンマ関数との関係
\[\gamma\left(a,x\right)+\Gamma\left(a,x\right)=\Gamma\left(a\right)\]
第1種・第2種不完全ガンマ関数の定義
\[\Gamma\left(a,x\right)=\int_{x}^{\infty}t^{a-1}e^{-t}dt\]
階乗と階乗の逆数の母関数
\[\frac{x^{a}}{a!}=e^{x}\left(\frac{\Gamma\left(a+1,x\right)}{\Gamma\left(a+1\right)}-\frac{\Gamma\left(a,x\right)}{\Gamma\left(a\right)}\right)\]
階乗冪(上昇階乗・下降階乗)の母関数
\[\sum_{k=0}^{\infty}P(k,n)x^{k}=\frac{x^{n}n!}{(1-x)^{n+1}}\]
ウォリス積分の拡張2重階乗表示
\[\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}}\]
1次式の総乗と階乗
\[\prod_{k=a}^{b}\left(kn+r\right)=n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)}\]
階乗の多重階乗表示
\[n!=\prod_{k=0}^{j-1}\left(n-k\right)!_{j}\]
多重階乗の階乗表示
\[\left(qn+r\right)!_{n}=r!_{n}n^{q}\frac{\left(q+\frac{r}{n}\right)!}{\left(\frac{r}{n}\right)!}\]
負の多重階乗
\[\left(-\left(qn+r\right)\right)!_{n}=\frac{\left(-1\right)^{q}}{\left(qn-\left(n-r\right)\right)!_{n}}\]
拡張多重階乗の簡単な値
\[0!^{n}=\frac{1}{\sqrt[n]{n}\left(\frac{1}{n}\right)!}\]