ベルヌーイ多項式

\[B_{n}\left(x\right)=\frac{D}{e^{D}-1}x^{n}\]

\[P\left(x,n+1\right)=\sum_{k=0}^{n}\frac{n+1}{k+1}S_{1}\left(n,k\right)\left(B_{k+1}\left(x\right)-B_{k+1}\right)\]

\[\sum_{k=0}^{\infty}B_{k}\left(x\right)\frac{t^{k}}{k!}=\frac{te^{xt}}{e^{t}-1}\]

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

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

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

\[B_{n}\left(0\right)=B_{n}\]

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

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