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I want to know if there is a polynomial formula for this (in general): Given $f(x)=\sum _{i=0}^n a_ix^i$ where $a_i, x^i , n \in \mathbb N^* $ . Given $f_1(n) = f(x)$, we define recursively $f_{k+1}(n)= \sum _{i=1}^n f_k(i)$. Is there a general polynomial formula for $f_{k+1}(n)$ ? Suppose $ f(x) = m$ for all $m \in \mathbb N^*$. Then we have $f_{k}(n)= mn^k$. If $f(x)=\sum _{i=0}^1 a_ix^i$, then $f_{k} (n) = a _1 \binom {n+k}{k+1}+ a_0 n^k$. Thank you.

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I'm note sure I'm following. When you write "$f(x) = m$ for all $m \in \mathbb{N}$", do you mean "$x \in \mathbb{N}$" instead? If so, shouldn't you have $f_2(n) = nm$ and $f_3(n) = \sum_{i= 1}^n km = \frac{n(n+1)}{2} m$ ? –  Joel Cohen Nov 25 '11 at 2:54
    
@Cohen: No Sir, m represents a constant polynomial where m is any natural number or zero. –  Keneth Adrian Nov 25 '11 at 9:33
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