Question

一道关于范德蒙德行列式的问题如图,这样一个范德蒙德行列式构成的函数,我想问下这个范德蒙德行列式的x,x2,x3,x4的余

Answer 1

f(x) = a15A15+a25A25+a35A35+a45A45+a55A55

= A15+xA25+x^2A35+x^3A45+x^4A55

= M15 - xM25 + x^2M35 - x^3M45 + x^4M55

而 f(x) = (x-a)(x-b)(x-c)(x-d) (b-a)(c-a)(d-a)(c-b)(d-b)(d-c)

= abcd(b-a)(c-a)(d-a)(c-b)(d-b)(d-c)

- (abc+abd+adc+bcd)(b-a)(c-a)(d-a)(c-b)(d-b)(d-c) x

+ (ab+ac+ad+bc+bd+cd)(b-a)(c-a)(d-a)(c-b)(d-b)(d-c) x^2

-(a+b+c+d)(b-a)(c-a)(d-a)(c-b)(d-b)(d-c) x^3

+(b-a)(c-a)(d-a)(c-b)(d-b)(d-c) x^4

比较系数,即得结论