3

Let $p$ be a polynomial. Let

$$S_p(n)=\sum_{k=1}^n p(k)$$

be the sum of the polynomial's values at the first $n$ integers.

Splitting the sum along each term's monomials, pulling the common coefficients before the sums and applying Faulhaber's formula yields result that

  1. $S_p$ is a polynomial in $n$ and that

  2. $\deg(S_p) = \deg(p) + 1$

This seems rather simple, so my question is: Do we really need to use the relatively heavy machinery of Faulhaber (giving exact coefficients) here or is there a way to immediately see that these two points are true?

  • 1
    writing $p(x)$ as a linear combination of $\binom{x}{m}$'s rather than of $x^m$'s makes it simpler – user8268 Dec 30 '14 at 10:06
  • All you are using (by linearity as you noted yourself) is that $\sum_{k=1}^n k^p$ is a polynomial in $n$ of degree $p+1$. This is shown here http://math.stackexchange.com/questions/18983/why-is-sum-limits-k-1n-km-a-polynomial-with-degree-m1-in-n?rq=1 – PhoemueX Dec 30 '14 at 16:50
  • Linked. – Alex Ravsky Aug 18 '17 at 09:31

0 Answers0