Exercise 6, § 6

Algebra, IV, p. 99

Let $n$ be an integer $\geq 1$ , and let $X_1, \ldots, X_n$ be indeterminates. For $1 \leq j \leq n$ let $s_j$ be the elementary symmetric polynomial of degree $j$ in the $X_i$ and $u_j = (-1)^{j+1} s_j$ . Put

S(T) = \prod_{i=1}^n (1 - X_i T) = 1 - \sum_{j=1}^n u_j T^j

and for each $k \geq 1$ put $p_k = \sum_{i=1}^n X_i^k$ ; show that

p_k = \sum a_{\lambda_1 \lambda_2 \ldots \lambda_n} u_1^{\lambda_1} u_2^{\lambda_2} \ldots u_n^{\lambda_n}

the summation being extended to all systems $(A_i, \ldots, A_i)$ of integers $\geq 0$ such that $\lambda_1 + 2\lambda_2 + \ldots + n\lambda_n = k$ , and the coefficient $a_{\lambda_1 \lambda_2 \ldots \lambda_n}$ being equal to

\frac{k(\lambda_1 + \lambda_2 + \cdots + \lambda_n - 1)!}{\lambda_1! \lambda_2! \ldots A_n!}

(expand the series $S'(T)/S(T)$ in a formal power series in $T$ ).

Solution

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