For context, I'm an A Level student (16 years old) but I have a passion for mathematics and do a lot of it in my spare time.
I found an interesting question in an A Level textbook a while ago that asked for polynomials $f_2(r), f_3(r)$, and $f_4(r)$ such that $\sum\limits_{r=1}^{n}f_2(r) = n^2$, $\sum\limits_{r=1}^{n}f_3(r) = n^3$, and $\sum\limits_{r=1}^{n}f_4(r) = n^4$.
I found these polynomials and then pushed myself to find a general formula for $f_a(r)\ \forall\ a \in \mathbb{Z}^+$. This formula turns out to be $f_a(r) = r^a - (r-1)^a$ and I proved that $\sum\limits_{r=1}^{n}f_a(r) = n^a\ \forall\ a, n \in \mathbb{Z}^+,\ n > 1$ in a paper that I can provide if requested.
My question here is, does this function have a name in this context of summation? I'd love to know more about this but I don't really know what to Google. For now, I've named it after myself as a bit of fun, but I'd love to know if it's got a proper name, and if it would actually be useful anywhere.
If $\phi(r)$ is a function defined on integers, then the (backward) difference is $\psi(r) = \phi(r) -\phi(r-1)$. To go from the difference to the original is (indefinite) summation. If $\phi(n) = \sum_{r=1}^n \psi(r)$ then the difference of $\phi$ is $\psi$. See https://en.wikipedia.org/wiki/Finite_difference – GEdgar Dec 31 '21 at 20:56