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F. C.
F. C.
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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

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.

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

Source Link
F. C.
F. C.
  • 3.6k
  • 20
  • 30

Is there an oscillating analog of the Gaussian distribution?

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.