I've been experimenting with the summation of polynomials. My line of attack is to treat the subject the way I would for calculus, but not using limits.
By way of a very simple example, suppose I wish to add the all numbers between $10$ and $20$ inclusive, and find a polynomial which I can plug the numbers into to get my answer. I suspect its some form of polynomial with degree $2$. So I do a integer 'differentiation': $$ \mathrm{diff}\left(x^{2}\right)=x^{2}-\left(x-1\right)^{2}=2x-1 $$
I can see from this that I nearly have my answer, so assuming an inverse 'integration' operation and re-arranging: $$ \frac{1}{2}\mathrm{diff}\left(x^{2}+\mathrm{int}\left(1\right)\right)=x $$
Now, I know that the 'indefinite integral' of 1 is just x, from 'differentiating' $x-(x-1) = 1$. So ultimately: $$ \frac{1}{2}\left(x^{2}+x\right)=\mathrm{int}\left(x\right) $$
So to get my answer I take the 'definite' integral: $$ \mathrm{int}\left(x\right):10,20=\frac{1}{2}\left(20^{2}+20\right)-\frac{1}{2}\left(9^{2}+9\right)=165 $$ (the lower bound needs decreasing by one)
My question is, is there a general way I can 'integrate' any polynomial, in this way?
Please excuse my lack of rigour and the odd notation.