How to evaluate / approximate polynomial sum expression over integers? Given
$$P(x) = \sum_{k=0}^Nc_kx^k$$
Can we derive some exact formula for
$$\sum_{k=0}^N P(k)$$
The only thing I can think of is the integral approximation:
$$\int_{0}^{N} P(x)dx \approx \sum_{k=0}^N P(k)$$
or maybe broken into smaller pieces
$$\int_{k}^{k+1} P(x)dx \approx \frac{P(k)+P(k+1)} 2$$
Maybe there are better approaches?
 A: So you are looking for a "nice" form to express
$$
\sum\limits_{j = 0}^n {P(j)}  = \sum\limits_{k = 0}^n {c_{\,k} \sum\limits_{l = 0}^n {l^{\,k} } } 
$$
Now, as far as I know, the different formulations for the sums of powers can be summarized in
$$
\eqalign{
  & S_m (n + 1) = \sum\limits_{0\, \le \,l\, \le \,n} {l^{\,k} } \quad \left| {\;0 \le {\rm integer }k,n} \right. =   \cr 
  &  = \sum\limits_{\left( {0\, \le } \right)\,j\,\left( { \le \,k} \right)} {\left\langle \matrix{
  k \cr 
  j \cr}  \right\rangle \left( \matrix{
  n + 1 + j \cr 
  k + 1 \cr}  \right)}  = \sum\limits_{\left( {0\, \le } \right)\,j\,\left( { \le \,k} \right)} {\;j!\;\left\{ \matrix{
  k \cr 
  j \cr}  \right\}\left( \matrix{
  n + 1 \cr 
  j + 1 \cr}  \right)}  =   \cr 
  &  = {1 \over {k + 1}}\sum\limits_{0\, \le \,j\, \le \,m} {\left( \matrix{
  k + 1 \cr 
  j \cr}  \right)\;B(j)\;\left( {n + 1} \right)^{\,k + 1 - j} }  \cr} 
$$
where:
 - the angle brackets denote the Eulerian Numbers;
 - the curly brackets denote the Stirling Numbers of 2nd kind;
 - $B(j)$ denote the Bernoulli Numbers.
Therefore, you cannot find a close form in general, unless the $c_k$ obey to a particular  function of the index.
A: $$ h \sum_{k=0}^{N-1} P(kh) > \int_{0}^{1} P(x)dx > h\sum_{k=1}^N P(kh)$$ 
where $h=1/N$. 
As N becomes larger, the magnitude of difference of the terms in the inequality decreases.
