Prove that $h_r(x_1,\dots,x_n)=\sum^n_{k=1}x^{n-1+r}_k\prod_{i\neq k}(x_k-x_i)^{-1}$

Question

Let $h_r\left(x_1,\ldots,x_n\right)$ denote the $r$-th complete homogeneous symmetric polynomial in the $n$ indeterminates $x_1, \ldots, x_n$ (that is, the sum of all degree-$r$ monomials in these $n$ indeterminates). How do I show that

$$h_r(x_1,\dots,x_n)=\sum^n_{k=1}x^{n-1+r}_k\prod_{i\neq k}(x_k-x_i)^{-1}$$

Can anyone just give me like a hint or "headstart"? Thanks!

Answer 1

Hint: Use the formula $s_{\lambda}=\frac{a_{\lambda+\delta}}{a_{\delta}}$ then expand its determinant.