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.
A composition on finite integral sequence
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sequences-and-series
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0@Cohen: No Sir, m represents a constant polynomial where m is any natural number or zero. – 2011-11-25