差商
, x n ] ⋅ g [ x n ] {\displaystyle (f\cdot g)[x_{0},\dots ,x_{n}]=f[x_{0}]\cdot g[x_{0},\dots ,x_{n}]+f[x_{0},x_{1}]\cdot g[x_{1},\dots ,x_{n}]+\dots +f[x_{0},\dots ,x_{n}]\cdot g[x_{n}]}
対称性: f [ x 0 , ... , x n ] = f [ x σ ( 0 ) , ... , x σ ( n ) ] ( σ ∈ S n ) {\displaystyle f[x_{0},\dots ,x_{n}]=f[x_{\sigma (0)},\dots ,x_{\sigma (n)}]\quad (\sigma \in S_{n})}
差商に対する平均値の定理より: f [ x 0 , ... , x n ] = f ( n ) ( ξ ) n ! ( ∃ ξ ∈ ( min x k , max x k ) ) {\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}\quad (\exists \xi \in (\min x_{k},\max x_{k}))}