I was trying to prove that given $$ a_n=\sum_{k=1}^m b_k c_k^n $$ then $a_n$ could be expressed by the recursive formula $$ a_n=-\sum_{k=1}^m u_ka_{n-k} $$ where $\prod_{j=1}^m(X-c_j)=X^m+\sum_{j=1}^n u_jX^{m-j}$
I believe that it follows from the Newton identities but I don't know how to proceed.
Hint: Then Newton's identities can be stated as
$$ke_{k}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{k}(-1)^{i-1}e_{k-i}(x_{1},\ldots ,x_{n})p_{i}(x_{1},\ldots ,x_{n}),$$ valid for all n ≥ 1 and n ≥k ≥ 1.