ミクシンスキーの演算子法
可微分な f に対して s f = f ′ + [ f ( 0 ) ] . {\displaystyle sf=f'+[f(0)].} s n f = f ( n ) + [ f ( n − 1 ) ( 0 ) ] + [ s f ( n − 2 ) ( 0 ) ] + ⋯ + [ s n − 1 f ( 0 ) ] . {\displaystyle s^{n}f=f^{(n)}+[f^{(n-1)}(0)]+[sf^{(n-2)}(0)]+\cdots +[s^{n-1}f(0)].} s { e α x } = [ α ] { e α x } + [ 1 ] , ( α ∈ C ) . {\displaystyle s\left\{e^{\alpha x}\right\}=[\alpha ]\left\{e^{\alpha x}\right\}+[1],(\alpha \in \mathbb {C} ).} s { x n } = [ n ] { x n − 1 } . {\displaystyle s\left\{x^{n}\right\}=[n]\left\{x^{n-1}\right\}.}
1 ( s − [ α ] ) n = { x n − 1 e α x ( n − 1 ) ! } . {\displaystyle {\frac {1}{(s-[\alpha ])^{n}}}=\left\{{\frac {x^{n-1}e^{\alpha x}}{(n-1)!}}\right\}.}