オイラー多項式

\[E_{n}\left(x\right)=\frac{2}{e^{D}+1}x_{n}\]

\[E_{n-1}\left(x\right)=\frac{2}{n}\sum_{k=0}^{n}C\left(n,k\right)\left(1-2^{k}\right)B_{k}x^{n-k}\]

\[\sum_{k=0}^{\infty}\frac{E_{k}\left(x\right)}{k!}t^{k}=\frac{2e^{xt}}{e^{t}+1}\]

\[E_{n}^{\left(k\right)}\left(x\right)=P\left(n,k\right)E_{n-k}\left(x\right)\]

\[E_{n}\left(x+y\right)=\sum_{k=0}^{n}C\left(n,k\right)E_{k}\left(x\right)y^{n-k}\]

\[E_{n}\left(1-x\right)=\left(-1\right)^{n}E_{n}\left(x\right)\]

\[E_{n}\left(\frac{1}{2}\right)=\frac{E_{n}}{2^{n}}\]

\[E_{n}\left(x\right)=\sum_{k=0}^{n}C\left(n,k\right)\frac{E_{k}}{2^{k}}\left(x-\frac{1}{2}\right)^{n-k}\]