Question

【线性代数】计算f(x+1）-f(x)

Answer 1

f(x+1)-f(x)=
1 0 0 0 ... 0 (x+1) - x
1 2 0 0 ... 0 (x+1)^2 - x^2
1 3 3 0 ... 0 (x+1)^3 - x^3
... ...
1 n C(n,2) C(n,3) ... C(n,n-1) (x+1)^n - x^n
1 n+1 C(n+1,2) C(n+1,3) ... C(n+1,n-1) (x+1)^(n+1)- x^(n+1)
第n+1列减第1列, 减x倍第2列, 减x^2第3列, ... , 减 x^(n-1) 倍第n列
1 0 0 0 ... 0 0
1 2 0 0 ... 0 0
1 3 3 0 ... 0 0
... ...
1 n C(n,2) C(n,3) ... C(n,n-1) 0
1 n+1 C(n+1,2) C(n+1,3) ... C(n+1,n-1) C(n+1,n)x^n
= n! C(n+1,n)x^n
= (n+1)! x^n.