The relationship between $f_{i}$

Question

Let $\lambda_{i}$ be a real number for any $1 \le i \le n$, defining $f_{i}$ as following $$f_{i} = \sum^{n}_{j=1} (\lambda_{j})^{i}$$ Suppose we know $f_{1},f_{2},f_{3}$ and $f_{4}$, then can we express $f_{5}$ and $f_{6}$ by $f_{1},f_{2},f_{3}$ and $f_{4}$? If yes, what the formulas in detail? Is there any references and books about it? I would be truly grateful for anyone who tells me anything about this question.

Answer 1

Let the $\lambda_j$ be the roots of the polynomial $\sum_{k=0}^n a_k\lambda^{n-k}$, with $a_0 = 1$. By Vieta's formulas, the $a_k$ are elementary symmetric polynomials in the $\lambda_j$ times $(-1)^k$. Then by Newton's identities, we have for $i \le n$ $$ f_i = -i a_i-\sum_{k = 1}^{i-1}a_k f_{i-k} $$ and for $i > n$, $$ f_i = -\sum_{k = 1}^na_{k}f_{i-k}. $$ Thus, if we know $f_1$, $f_2$, ..., $f_n$, we can reverse the first identity to get $a_k = -\sum_{i =1}^kf_i a_{k-i}/k$, and thus recursively find the $a_k$ in terms of the $f_i$. Then we can use the second identity recursively to find any $f_i$ for $i > n$ in terms of $f_1$, $f_2$, ..., $f_n$. Note that we need all $n$ values of $f$, as otherwise we can't pin down every value of $a_k$.

I'm not aware of any explicit formula for the $f_i$, but you'll find they are linear combinations of terms of the form $\prod_{j=1}^nf_j^{p_i}$, where $\sum jp_j = i$.