$S_k$ polinomu $k$. dereceden $n$ degiskenli elementer simetrik polinom ise $S_k(x+y)-S_k(x)-S_k(y)$

Question

$S_k$ polinomu $k$. dereceden $n$ degiskenli elementer simetrik polinom olsun.Sadece $x,y,S_1,S_2,\cdots S_{k-1}$'yi kullanarak nasil  $$S_k(x+y)-S_k(x)-S_k(y)$$ polinomunu yazabilirim, burda $x=(x_1,x_2,\cdots,x_n)$ ve $y=(y_1,y_2,\cdots,y_n)$.

ornek: $x=(x_1,x_2)$ ve $y=(y_1,y_2)$ olsun:

$$S_2(x+y)-S_2(x)-S_2(y)=(x_1+y_1)(x_2+y_2)-x_1x_2-y_1y_2$$ $$=x_1y_2+x_2y_1=(x_1+x_2)(y_1+y_2)-(x_1y_1+x_2y_2)$$ $$=S_1(x)S_1(y)-S_1(xy).$$

Bunu $n$ ve $k$ cinsinden nasil genellestirebiliriz?

Comments

http://en.wikipedia.org/wiki/Elementary_symmetric_polynomial