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Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows:

$p_0 = a_0$

$p_1 = a_0x+a_1$

$p_2 = a_0x^2+a_1x+a_2$

$p_n = a_0x^n+a_1x^{n-1}+...+a_n$

...

Where the coefficients are the elements of the sequence. The characteristics of $p_n$ are interesting because $p_n(0)=a_n$ and $p_n(1)=\sum_{n}a_n$

For instance, applied to the Möebius function:

$p_0 = 1$

$p_1 = x-1$

$p_2 = x^2-x-1$

$p_3 = x^3-x^2-x$

$p_4 = x^4-x^3-x^2-1$

...

$p_n(0)=\mu(n)$ ($n^{th}$ element of the Möbius sequence) and $p_n(1)=\sum_{n}\mu(n)$ is the partial summation up to $n$ of the Möbius function, in other words, it is the Merten's function.

This is the graph of the first $100$ polynomials, $p_0$ to $p_{99}$

enter image description here

And this is a zoom of the segment [0,1] so it is possible to see the ramifications of each polynomial from the position of the last $\mu(n)$ to the position of the last $\sum_n\mu(n)$. The shape of the paths is quite curious because it is a representation of the surjective-only diagram of $\mu(n) \to \sum_n \mu(n)$ for each $n$ in $[0,100]$.

enter image description here

I would like to ask the following questions:

  1. Is it possible to know something about the properties (e.g. the convergence of the accumulated sum) of a sequence as the $lim_{n \to \infty}p_n(1)=lim_{n \to \infty}\sum_{n}a_n$ by the calculation of the shape of the generic polynomial generated with the elements of the sequence as coefficients? e.g. finding the the shape of the "limit" polynomial $p_n$ when $n \to \infty$?

  2. I tried to find some papers about this kind of approach, to understand if it leads to something or it is just visually interesting. Are there any papers regarding the generation of polynomials by using the elements of sequences as coefficients? Thank you!

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  • $\begingroup$ The wording of your question is weirdly reminiscent of some thoughts I've been having recently, but I'm not sure it's exactly the same thing. $\endgroup$ – user301988 May 17 '16 at 1:12
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    $\begingroup$ Typically, one defines the generating function of such a sequence as $A(x)=\sum_{k=0}^\infty a_k x^k$. This in turn corresponds to a sequence of partial sums $A_n(x)=\sum_{k=0}^n a_k x^k$, and these relate to your polynomials as $p_n(x)=x^n A_n(x^{-1})$. $\endgroup$ – Semiclassical May 17 '16 at 1:53
  • $\begingroup$ @Semiclassical thank you very much for the insight. I did not know the correct name. $\endgroup$ – iadvd May 17 '16 at 2:47
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This really only addresses question 2, since the construction is not exactly what you describe - in particular, the indexing is reversed - but you may be interested in generating functions https://en.wikipedia.org/wiki/Generating_function.

Given a sequence $\mathcal{A}=(a_i)_{i\in\mathbb{N}}$, we can associate to it the formal power series $$a_0x^0+a_1x^1+a_2x^2+...$$ (this is the ordinary generating function). There are many other kinds of generating function we can associate to $\mathcal{A}$ - e.g. the exponential generating function $$\sum_{i=0}^\infty {a_ix^i\over i!}.$$

Often this formal power series will actually be equal (on some open neighborhood) to some function $f$, and by studying $f$ it turns out we can indeed gain information about $\mathcal{A}$. I'm going to stop here, because

  • there is a truly gargantuan amount of research about generating functions, and

  • I don't know any of it,

but hopefully you find this valuable (and hopefully someone who actually knows things about generating functions stops by to give a better answer!).

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    $\begingroup$ The main references I'd suggest on generating functions are Wilf's generatingfunctionology and Flajolet & Sedgwick's Analytic Combinatorics. The latter is particularly relevant insofar as it addresses quite a bit of the asymptotic analysis of GFs. Both are available on their authors' websites in pdf form. $\endgroup$ – Semiclassical May 17 '16 at 1:57
  • $\begingroup$ @NoahSchweber thank you for the explanation! I did not know that is is referred as "generating functions". I was also thinking about the construction of the coefficients in the opposite way, but for some reason I thought that the initial terms are the ones with more weight (because the are from the beginning) in the evolution of the sequence, so they should be the coefficients of the greater degrees and that was wrong. $\endgroup$ – iadvd May 17 '16 at 2:46
  • $\begingroup$ @NoahScheweber oh now I remember why I decided that the last element $a_n$ added to $p_n$ would be the last coefficient of the polynomial (the free coefficient of degree $x^0$)! It is because by doing so it is possible to obtain $a_n$ at $p_n(0)$ while it is not possible if the standard definition is used. So I can plot the graph of the relationship of $a_n$ with $\sum_n a_n$ like in the examples of the question. $\endgroup$ – iadvd May 17 '16 at 3:02

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