Coefficients of Polynomials Corresponding to Sums of Powers of Natural Numbers Sum to 1.

September 6, 2012

This post has a pretty weird title, but the problem is easy to state and uses a few interesting mathematical concepts.  It’s worth going through.  Let’s start with the basics.

 

Problem 1.  Let S_{\alpha}(n) = 1^{\alpha} + 2^{\alpha} + \dots + n^{\alpha}.  Show that S_{\alpha}(n) is a polynomial for each \alpha\in {\mathbb N} and that the degree of the polynomial is \alpha + 1.

 

Indeed, for example, we have  that 1 + 2 + \dots + n = \frac{n(n+1)}{2}, as we learned in Calculus, and this is a polynomial of degree 2.  Similarly, 1^{2} + 2^{2} + \dots + n^{2} = \frac{n(n+1)(2n+1)}{6}, which is a polynomial of degree 3.  In the same respect, 1^{3} + 2^{3} + \dots + n^{3} = \frac{(n(n+1))^{2}}{4}, which is a polynomial of degree 4.

The associated polynomials in this case are given by Faulhaber’s formula:

 

Theorem (Faulhaber).  For \alpha\in {\mathbb N} we have \displaystyle \sum_{k=1}^{n}k^{\alpha} = \frac{1}{1 + \alpha}\sum_{j=0}^{\alpha}(-1)^{j}\binom{\alpha + 1}{j}B_{j}n^{\alpha + 1 - j}.

 

This formula looks terrifying, but it is not hard to apply in practice.  You may be wondering, though, what the B_{j}‘s in this formula stand for.  These are the strange and wonderful Bernoulli numbers, of course!  I always enjoy seeing these creatures, because they unexpectedly pop up in the strangest problems.  There are a number of ways to define these numbers, one of which is to just write them out sequentially, starting with B_{0}:

 

1, -\frac{1}{2}, \frac{1}{6}, 0, -\frac{1}{30}, 0, \frac{1}{42}, 0, -\frac{1}{30}, 0, \dots

 

But in this case it is not so easy to guess the next value.  The clever reader will notice that all of the odd numbered Bernoulli numbers (except the first) are zero, but other than that there does not seem to be a clear pattern.  Fortunately, we can construct a function which generates the values as coefficients; we’ll call this function (surprise!) a generating function.

 

Definition.  We define the sequence \{B_{j}\}_{j=0}^{\infty} by

\displaystyle \frac{t}{e^{t} - 1} = \sum_{j=0}^{\infty}B_{j}\frac{t^j}{j!}.

 

Notice that this will, in fact, generate the B_{j} as coefficients times \frac{1}{j!}.  Neat.  In practice, you can use a program like Mathematica to compute B_{j} for pretty large values of j; but, of course, there are lists available.  We can now use Faulhaber’s formula above, which gives us (assuming we have proven that the formula holds!) that the sums of powers of natural numbers form polynomials of degree \alpha + 1.

 

But something else happens that’s pretty interesting.  Let’s look at some of the functions.

 

\begin{array}{|c|c|}\hline \alpha & \mbox{Polynomial}\\\hline &\\ 1 & \frac{1}{2}n^{2} + \frac{1}{2}n\\ &\\ 2 & \frac{1}{3}n^{3} + \frac{1}{2}n^{2} + \frac{1}{6}n\\ & \\ 3 & \frac{1}{4}n^{4} + \frac{1}{2}n^{3} + \frac{1}{4}n^{2} \\ & \\ 4 & \frac{1}{5}n^{5} + \frac{1}{2}n^{4} + \frac{1}{3}n^{3} - \frac{1}{30}n\\ & \\ \hline \end{array}

 

Look at the coefficients in each of these polynomials.  Anything strange about them?  Consider them for a bit.

 

Problem.  Look at the coefficients.  What do you find interesting about them?  Note that, in particular, for a fixed \alpha, the coefficients of the associated polynomial sum to 1.  Convince yourself that this is probably true (do some examples!) and then prove that it is true.  Do this before reading the statements below.

 

Anecdote.  I spent quite a while trying to write down the "general form" of a polynomial with elementary symmetric polynomials and roots to try to see if I could prove this fact using some complex analysis and a lot of terms canceling out.  This morning, I went into the office of the professor to ask him about what it means that these coefficients sum up to 1.  He then gave me a one-line (maybe a half-line) proof of why this is the case. 

 

Hint.  What value would we plug in to a polynomial to find the sum of the coefficients?  What does plugging in this value mean in terms of the sum?

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One Response to “Coefficients of Polynomials Corresponding to Sums of Powers of Natural Numbers Sum to 1.”

  1. j2kun said

    Nice post. You may enjoy a sort of geometric proof of a special case of this theorem. See my entry: http://jeremykun.wordpress.com/2011/06/24/sums-of-the-first-n-numbers-squares/

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