01 March 2008

Zeros of some polynomials arising from sums

Here's a little thing I thought of a few days ago. Consider the following identities for the sums of powers:
\sum_{k=1}^n k^0 = n

(okay, that's kind of stupid, but you have to start somewhere...),
\sum_{k=1}^n k^1 = {n(n+1)\over 2};

\sum_{k=1}^n k^2 = {n(n+1)(n+1/2)\over 3}

(the right-hand side might be more familiar as n(n+1)(2n+1)/6), and
\sum_{k=1}^n k^3 = {n^2(n+1)^2\over 4}

(the right-hand side here is,coincidentally the square of (1+2+...+k). For each choice of exponent we get a different polynomial in the numerator. They all factor into linear terms... that doesn't keep up, though. For example,
\sum_{k=1}^n k^9 = {n^2(n^2+n-1)(n+1)^2 (n^4+2n^3 - n^2/2 - 3n/2 + 3) \over 10}

Still, one wonders -- what are the roots of these polynomials? (The first thought is that they're always in the interval [-1, 0], but that's pretty quickly disproven by considering the sum of 5th powers.)

Some computation shows that the patterns of zeroes in the complex plane are both symmetric around the real axis (no surprise there; zeroes come in complex conjugate pairs!) and around the line y = -1/2 (a bit more surprising). So you think to plot them, and you get something that looks like this plot for the polynomial you obtain when you sum 300th powers. (I didn't make that plot; it's from Richard Stanley's web page on interesting zeros of polynomials.)

It turns out that they're the Bernoulli polynomials; for very large n Veselov and Ward showed that the real zeroes are very near 0, ± 1/2, ± 1, ... if n is odd, and ± 1/4, ± 3/4, ± 5/4, ... if n is even; furthermore, in the limit, the nth Bernoulli polynomial has 2/(πe)n real zeros. (2/πe is about .235; thus in the 300th Bernoulli polynomial we expect about 70 real zeros, taking up an interval of length 35 or so centered at -1/2 on the real line; that's what you see in that plot.)

Goh and Boyer (who I've mentioned before for similar work on partition polynomials) have found the "zero attractor" of the Euler polynomials, and state in their paper that the methods there also give a similar result for the Bernoulli polynomials -- basically, what this means is that if we shrink down the plot of the zeros of the nth Bernoulli polynomial by a factor of n, then the zeroes fall very close to some limiting curves and are arranged with a certain density along those curves. (Along the portion of the real axis in question, the density is constant; along the other branches it doesn't seem to be.)

William M. Y. Goh, Robert Boyer. On the Zero Attractor of the Euler Polynomials. arXiv: math.CO/0409062. (2004)
Alexander Veselov and Joseph Ward, On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials, arXiv: math.GM/0205183. (2002)

No comments: