3
$\begingroup$

Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $E^{(k)}(z)$ correspond to left-shifts (and truncation) of the original sequence.

Something odd happens if one visualizes the sequence of functions $E^{(k)}(z)$ (for some "typical" sequence $a_n$, to be explained) as an animated movie, with $k$ being the time variable: the animation appears to be "smooth" (visually) and that there is a "black hole" at the origin, hoovering up the function, frame by frame.

To anchor what might otherwise seem like a screwball question, a reference animation showing the effect can be found here: https://linas.org/art-gallery/egf/rand-exp-deriv-ani.gif

By "hoovering up" I mean this: If one plots the zeros of $E^{(k)}(z)$, one can watch them slowly, gradually shift towards the origin; a single zero disappears at the origin, each frame (of course; the "strength" is reduced with each shift). What is surprising is that there is a visually smooth flow; one might have expected the locations of the zeros to bounce around unpredictably, but no; there is a clear continuity as each zero drifts towards the origin. The drift is not precisely on a ray, of course, but it does seem to be irrotational, and always inward. There is no visual ambiguity as to which zero is which: you get the sense of tracking the "same" zero, from frame to frame. (On rare occasions, you can see two zeros merge on the negative real axis, and then split again, with both then falling in radially, one faster than the other.)

The sequences that I've looked at so far are random samplings of integers from a space $1<a_n\le n$, but I think there's nothing particularly special about this sample space; it seems that the above qualitative behavior would persist if one picked reals, or even complex numbers, with other distributions. The only nice thing about sampling from $1<a_n\le n$ is that one gets a nice distribution of zeros scattered all over the complex plane, and the resulting generating function seems to usually be entire (i.e. lacking in poles).

Of course, there are specific choices of $a_n$ that are counter-examples to everything I claim; I'm interested in the "almost everywhere" case. The sampling space $1<a_n\le n$ means that these are not full shifts or subshifts, in the usual sense, but are "almost" so. The visual smoothness suggests that there should be an actual smoothness. But how?

The questions are these: is this a "known thing"? What is the structure of the sample space $\{a_n\}$ for which this occurs? Is there a natural extension of the visual flow to an actually smooth flow? Can one always assign consistent labels to the zeros, frame-by-frame? (is the vector field interpolable and integrable? Could the flow ever be geodesics of something?) Is the movement always infalling? Is the movement always "irrotational" (for a suitable definition of what "irrotational" means)?

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.