7
$\begingroup$

It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the Gaussian law. One obvious example is given by the familly $(1+x)^n$.

I know some examples of famillies of polynomials where the coefficients are not positive, but where plotting the coefficients give a nice oscillating curve, as the one displayed below.

Plot of coefficients of some large polynomial

My question is the following:

Is there any known such oscillating function, with some kind of universal property ?

In other words, what is the function one can see in this picture ?

As far as I can tell, this does not seem to be given by a product of $\exp(-x^2)$ by trigonometric functions, because the local maxima do not fit very well on a parabola.

EDIT:

Here is a graph of the log of the absolute value of the coefficients, compared with a parabola.

Comparison of log with parabola

$\endgroup$
9
  • 2
    $\begingroup$ The convergence to a Gaussian that you describe is essentially due to the central limit theorem, so you basically seem to be looking for a non-positive analog of a stable distribution. What is that picture of yours a plot of, anyway? $\endgroup$ – Ilmari Karonen Feb 24 '14 at 12:31
  • $\begingroup$ The picture is the plot of the list of coefficients of one polynomial (in a familly of polynomials indexed by the integers). This is essentially a sequence of points, one with coordinates $(i,c_i)$ for each monomial $c_i x^i$. $\endgroup$ – F. C. Feb 24 '14 at 14:54
  • $\begingroup$ Yes, I got that, I was just curious about what the specific family was. $\endgroup$ – Ilmari Karonen Feb 24 '14 at 14:55
  • $\begingroup$ The polynomials are numerators of $q$-Bernoulli polynomials (and variants of that). $\endgroup$ – F. C. Feb 24 '14 at 14:59
  • 2
    $\begingroup$ Oops, I should have said "numerators of $q$-Bernoulli numbers". These rational functions were introduced by Carlitz in Carlitz, L. q-Bernoulli numbers and polynomials. Duke Math. J. 15, (1948). 987–1000. They are available in sage as follows: "from sage.combinat.q_bernoulli import q_bernoulli" then "q_bernoulli(20).numerator()" $\endgroup$ – F. C. Feb 25 '14 at 8:19
5
$\begingroup$

Another ... the Hermite functions. Hermite polynomials $H_n(x)$ times the density $\exp(-x^2)$. Here is $H_{20}(x)\exp(-x^2)$

herm

$\endgroup$
2
  • $\begingroup$ This looks like a possible answer indeed. Sometimes I also see odd functions instead of even ones. They resemble to $H_{odd}(x) \exp(-x)$. $\endgroup$ – F. C. Feb 25 '14 at 11:21
  • $\begingroup$ I am now more skeptical about this proposal, because in my examples, the oscillations are slower in the middle, whereas they are faster in the middle for $H_n(x)exp(-x^2)$. $\endgroup$ – F. C. Mar 10 '16 at 10:58
3
$\begingroup$

Note that $(1+x)^n$ itself does not have Gaussian coefficients when you go that far from the central term: by Stirling, for fixed $\rho \in (0,1)$ the $x^{\rho n}$ coefficient $\bigl( {n \atop \rho n} \bigr)$ is roughly proportional to $\exp nH(\rho)$ where $H(\rho) = -\rho \log \rho - (1-\rho) \log(1-\rho)$. Your graph looks like it could be something like this multiplied by $\cos cx$ for some $c \approx 1/6$. The stationary-phase technique for finding the asymptotic behavior of power-series coefficients often gives rise to expressions such as this, though $H(r)$ might be a sum of more complicated terms than just $-\rho \log \rho$ and $-(1-\rho) \log(1-\rho)$.

$\endgroup$
0
$\begingroup$

I would guess sinc.${}{}{}{}{}{}{}{}{}{}{}$

$\endgroup$
1
  • 1
    $\begingroup$ It seems to me that the function $sin(x)/x$ is not decreasing fast enough. Its graph does not look similar, and has long tails and many visible oscillations. $\endgroup$ – F. C. Feb 24 '14 at 19:09
0
$\begingroup$

A nice one is the Dirichlet kernel.

Here is $D_5(x)$

D5

$\endgroup$
1
  • 2
    $\begingroup$ But that's just a finite approximation to sinc, which we already know does not decay fast enough to match the coefficients of the OP's polynomial (though it's certainly nice for other applications). $\endgroup$ – Noam D. Elkies Feb 24 '14 at 22:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.