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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

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    $\begingroup$ 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
    Commented Feb 24, 2014 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.
    Commented Feb 24, 2014 at 14:54
  • $\begingroup$ Yes, I got that, I was just curious about what the specific family was. $\endgroup$
    – Ilmari Karonen
    Commented Feb 24, 2014 at 14:55
  • $\begingroup$ The polynomials are numerators of $q$-Bernoulli polynomials (and variants of that). $\endgroup$
    – F. C.
    Commented Feb 24, 2014 at 14:59
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    $\begingroup$ 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.
    Commented Feb 25, 2014 at 8:19

4 Answers 4

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

herm

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  • $\begingroup$ 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.
    Commented Feb 25, 2014 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.
    Commented Mar 10, 2016 at 10:58
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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 $-(1-\rho) \log(1-\rho)$.

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

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    $\begingroup$ 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.
    Commented Feb 24, 2014 at 19:09
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A nice one is the Dirichlet kernel.

Here is $D_5(x)$

D5

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    $\begingroup$ 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
    Commented Feb 24, 2014 at 22:20

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